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Linear Algebra For Undergraduate Students

Yilan(Peter) Tan

Section 4.3 Abstract Vector Spaces

While \(\mathbb{R}^n\) provides a concrete and familiar setting for studying linear algebra, the concepts we have learned extend far beyond column vectors. Many mathematical objects can be viewed as vectors if they satisfy the fundamental properties of vector addition and scalar multiplication.
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Subsection 4.3.1 Definition of Vector Space

Before exploring specific examples of vector spaces, we need to formally define what a vector space is. The beauty of this definition is that it captures the essential algebraic structure shared by many different mathematical objects.
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Definition 4.3.1. Vector Space.

A vector space is a nonempty set \(V\) of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms listed below. The axioms must hold for all vectors \(\mathbf{u}\text{,}\) \(\mathbf{v}\text{,}\) and \(\mathbf{w}\) in \(V\) and for all scalars \(c\) and \(d\text{.}\)
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Axioms of Addition:
  1. Closure under addition: The sum of \(\mathbf{u}\) and \(\mathbf{v}\text{,}\) denoted by \(\mathbf{u}+\mathbf{v}\text{,}\) is in \(V\text{.}\)
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  2. Commutativity: \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\)
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  3. Associativity: \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\)
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  4. Zero vector: There is a zero vector \(\mathbf{0}\) in \(V\) such that \(\mathbf{u}+\mathbf{0}=\mathbf{u}\text{.}\)
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  5. Additive inverse: For each \(\mathbf{u}\) in \(V\text{,}\) there is a vector \(-\mathbf{u}\) in \(V\) such that \(\mathbf{u}+(-\mathbf{u})=\mathbf{0}\text{.}\)
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Axioms of Scalar Multiplication:
  1. Closure under scalar multiplication: The scalar multiple of \(\mathbf{u}\) by \(c\text{,}\) denoted by \(c\mathbf{u}\text{,}\) is in \(V\text{.}\)
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  2. Distributivity over vector addition: \(c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}\)
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  3. Distributivity over scalar addition: \((c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}\)
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  4. Compatibility of scalar multiplication: \(c(d\mathbf{u})=(cd)\mathbf{u}\)
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  5. Identity element of scalar multiplication: \(1\mathbf{u}=\mathbf{u}\)
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Remark 4.3.2.

These ten axioms divide naturally into two groups:
  • Axioms 1-5 describe properties of vector addition, ensuring that vectors can be added in a way consistent with our intuition from \(\mathbb{R}^n\text{.}\)
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  • Axioms 6-10 describe properties of scalar multiplication and how it interacts with vector addition.
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Notice that the definition is abstract: it doesn’t specify what the vectors "are" (numbers, polynomials, functions, etc.). Any set with two operations satisfying these axioms qualifies as a vector space. This generality is powerful—it allows us to apply the same theorems and techniques to many different mathematical contexts.
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Example 4.3.3. \(\mathbb{R}^n\) as a Vector Space.

The set \(\mathbb{R}^n\) with standard vector addition and scalar multiplication satisfies all ten axioms and is therefore a vector space. This is the prototypical example we have been working with throughout this chapter.
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Subsection 4.3.2 Examples of Vector Spaces

Definition 4.3.4. Polynomial Space \(\mathcal{P}_n(x)\).

Let \(\mathcal{P}_n(x)=\{a_0+a_1x+\ldots+a_nx^n|a_i\in \mathbb{R}\}\) be the set of all polynomials of degree at most \(n\) with real coefficients. This set forms a vector space with operations:
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Addition:
\begin{gather*} (a_0+a_1x+\ldots+a_nx^n)+(b_0+b_1x+\ldots+b_nx^n)\\ =(a_0+b_0)+(a_1+b_1)x+\ldots+(a_n+b_n)x^n \end{gather*}
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Scalar Multiplication:
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\begin{equation*} k(a_0+a_1x+\ldots+a_nx^n)=(ka_0)+(ka_1)x+\ldots+(ka_n)x^n \end{equation*}
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Definition 4.3.5. Matrix Space \(M_{m \times n}(\mathbb{R})\).

Let \(M_{m \times n}(\mathbb{R})=\left\{\left[ \begin{array}{ccc} a_{11} & \ldots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \ldots & a_{mn} \\ \end{array} \right]\middle| a_{ij}\in \mathbb{R} \right\}\) be the set of all \(m \times n\) matrices with real entries. This set forms a vector space with the usual matrix addition and scalar multiplication.
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Example 4.3.6.

For \(M_{2 \times 2}(\mathbb{R})\text{,}\) the standard basis consists of:
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\begin{equation*} E_{11} = \begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}, \quad E_{12} = \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}, \quad E_{21} = \begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}, \quad E_{22} = \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix} \end{equation*}
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Definition 4.3.7. Space of Continuous Functions.

Let \(C(-\infty, \infty)\) be the set of all real-valued continuous functions defined on the entire real line. This set forms a vector space with operations:
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Addition: \((f+g)(x)=f(x)+g(x)\)
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Scalar Multiplication: \((cf)(x)=c[f(x)]\)
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Definition 4.3.8. Space of Continuous Functions on an Interval.

Let \(C[a, b]\) be the set of all real-valued continuous functions defined on the closed interval \([a,b]\text{.}\) This set forms a vector space with the same operations as above.
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Example 4.3.9.

The functions \(f(x) = \sin(x)\text{,}\) \(g(x) = \cos(x)\text{,}\) and \(h(x) = e^x\) are all elements of \(C(-\infty, \infty)\text{.}\) The function \(2\sin(x) + 3\cos(x) - e^x\) is their linear combination in this vector space.
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All these examples share the fundamental properties that define a vector space:
  • Closure under addition and scalar multiplication
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  • Existence of a zero vector (zero polynomial, zero matrix, zero function)
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  • Existence of additive inverses
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  • Associativity and commutativity of addition
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  • Distributive properties of scalar multiplication
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Key Insight: The concepts of linear independence, span, basis, and dimension that we studied in \(\mathbb{R}^n\) apply equally well to these abstract vector spaces. This universality is what makes linear algebra such a powerful and widely applicable field of mathematics.
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Subsection 4.3.3 Basis of Abstract Vector Spaces

The concepts of span, linear independence, and basis that we developed for \(\mathbb{R}^n\) extend naturally to abstract vector spaces. These concepts are fundamental for understanding the structure of any vector space.
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Definition 4.3.10. Span.

Let \(V\) be a vector space and let \(S = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}\) be a set of vectors in \(V\text{.}\) The span of \(S\) is the set of all linear combinations of the vectors in \(S\text{:}\)
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\begin{equation*} \operatorname{span}(S) = \operatorname{span}\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k \mid c_i \in \mathbb{R}\}. \end{equation*}
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We say that \(S\) spans \(V\) if \(\operatorname{span}(S) = V\text{,}\) meaning every vector in \(V\) can be written as a linear combination of vectors in \(S\text{.}\)
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Example 4.3.11. Span in Polynomial Space.

In \(\mathcal{P}_2(x)\text{,}\) the set \(\{1, x, x^2\}\) spans the entire space because any polynomial \(a_0 + a_1x + a_2x^2\) can be written as \(a_0 \cdot 1 + a_1 \cdot x + a_2 \cdot x^2\text{.}\)
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Definition 4.3.12. Linear Independence.

A set of vectors \(S = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}\) in a vector space \(V\) is linearly independent if the only solution to the equation
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\begin{equation*} c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \end{equation*}
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is \(c_1 = c_2 = \cdots = c_k = 0\) (the trivial solution).
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If there exist scalars \(c_1, c_2, \ldots, c_k\text{,}\) not all zero, such that the equation holds, then \(S\) is linearly dependent.
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Example 4.3.13. Linear Independence in \(\mathcal{P}_2\).

Determine whether the set of vectors in \(\mathcal{P}_2\) is linearly independent or linearly dependent:
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\begin{equation*} S = \{1+x-2x^2, 2+5x-x^2, x+x^2\}. \end{equation*}
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Solution.
We need to determine if there exist scalars \(c_1, c_2, c_3\text{,}\) not all zero, such that
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\begin{equation*} c_1(1+x-2x^2) + c_2(2+5x-x^2) + c_3(x+x^2) = 0. \end{equation*}
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Expanding the left side:
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\begin{equation*} c_1 + c_1x - 2c_1x^2 + 2c_2 + 5c_2x - c_2x^2 + c_3x + c_3x^2 = 0. \end{equation*}
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Collecting terms by powers of \(x\text{:}\)
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\begin{equation*} (c_1 + 2c_2) + (c_1 + 5c_2 + c_3)x + (-2c_1 - c_2 + c_3)x^2 = 0. \end{equation*}
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Since this must equal the zero polynomial \(0 + 0x + 0x^2\text{,}\) we equate coefficients:
\begin{align*} c_1 + 2c_2 \amp= 0 \amp\amp \text{(constant term)}\\ c_1 + 5c_2 + c_3 \amp= 0 \amp\amp \text{(coefficient of } x\text{)}\\ -2c_1 - c_2 + c_3 \amp= 0 \amp\amp \text{(coefficient of } x^2\text{)} \end{align*}
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This gives us the augmented matrix:
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\begin{equation*} \left[\begin{array}{ccc|c} 1 \amp 2 \amp 0 \amp 0 \\ 1 \amp 5 \amp 1 \amp 0 \\ -2 \amp -1 \amp 1 \amp 0 \end{array}\right]. \end{equation*}
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Row reducing to RREF:
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\begin{equation*} \left[\begin{array}{ccc|c} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \end{array}\right]. \end{equation*}
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The only solution is \(c_1 = c_2 = c_3 = 0\) (the trivial solution). Therefore, \(S\) is linearly independent.
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Example 4.3.14. Linear Independence in \(M_{2,2}\).

Determine whether the set of vectors in \(M_{2,2}\) is linearly independent or linearly dependent:
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\begin{equation*} S = \left\{\begin{bmatrix}2 \amp 1\\0 \amp 1\end{bmatrix}, \begin{bmatrix}3 \amp 0\\2 \amp 1\end{bmatrix}, \begin{bmatrix}1 \amp 0\\2 \amp 0\end{bmatrix}\right\}. \end{equation*}
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Solution.
We need to determine if there exist scalars \(c_1, c_2, c_3\text{,}\) not all zero, such that
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\begin{equation*} c_1\begin{bmatrix}2 \amp 1\\0 \amp 1\end{bmatrix} + c_2\begin{bmatrix}3 \amp 0\\2 \amp 1\end{bmatrix} + c_3\begin{bmatrix}1 \amp 0\\2 \amp 0\end{bmatrix} = \begin{bmatrix}0 \amp 0\\0 \amp 0\end{bmatrix}. \end{equation*}
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Computing the left side:
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\begin{equation*} \begin{bmatrix}2c_1 + 3c_2 + c_3 \amp c_1\\2c_2 + 2c_3 \amp c_1 + c_2\end{bmatrix} = \begin{bmatrix}0 \amp 0\\0 \amp 0\end{bmatrix}. \end{equation*}
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Equating corresponding entries gives us the system:
\begin{align*} 2c_1 + 3c_2 + c_3 \amp= 0 \amp\amp \text{(entry (1,1))}\\ c_1 \amp= 0 \amp\amp \text{(entry (1,2))}\\ 2c_2 + 2c_3 \amp= 0 \amp\amp \text{(entry (2,1))}\\ c_1 + c_2 \amp= 0 \amp\amp \text{(entry (2,2))} \end{align*}
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From equation 2, \(c_1 = 0\text{.}\) Substituting into equation 4: \(0 + c_2 = 0\text{,}\) so \(c_2 = 0\text{.}\) From equation 3: \(2(0) + 2c_3 = 0\text{,}\) so \(c_3 = 0\text{.}\) We can verify this satisfies equation 1: \(2(0) + 3(0) + 0 = 0\) ✓
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The only solution is \(c_1 = c_2 = c_3 = 0\) (the trivial solution). Therefore, \(S\) is linearly independent.
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Definition 4.3.15. Basis.

A set of vectors \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n\}\) is a basis for a vector space \(V\) if:
  1. \(\mathcal{B}\) spans \(V\) (every vector in \(V\) can be expressed as a linear combination of vectors in \(\mathcal{B}\)), and
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  2. \(\mathcal{B}\) is linearly independent.
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Equivalently, \(\mathcal{B}\) is a basis if every vector in \(V\) can be expressed uniquely as a linear combination of vectors in \(\mathcal{B}\text{.}\)
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Proof.

The proof relies on the fact that if \(V\) has a basis of \(n\) vectors, then any set of more than \(n\) vectors in \(V\) must be linearly dependent, and any set of fewer than \(n\) linearly independent vectors cannot span \(V\text{.}\) This ensures that the number of vectors in a basis is an invariant property of the vector space, independent of which basis we choose.
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Definition 4.3.17. Dimension of a Vector Space.

Let \(V\) be a vector space with a basis \(\mathcal{B}\text{.}\) The dimension of \(V\text{,}\) denoted \(\dim(V)\text{,}\) is the number of vectors in the basis \(\mathcal{B}\text{.}\) If \(\dim(V) <\infty\text{,}\) the vector space is called finite-dimensional; otherwise, we say that \(V\) is infinite-dimensional.
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Example 4.3.18. Standard Basis for \(\mathcal{P}_n(x)\).

Standard Basis for \(\mathcal{P}_n(x)\text{:}\) The set \(\{1, x, x^2, \ldots, x^n\}\) forms a basis for \(\mathcal{P}_n(x)\text{.}\) This space has dimension \(n+1\text{.}\)
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Example 4.3.19. Standard Basis for \(M_{m\times n}(\mathbb{R})\).

Standard Basis for \(M_{m \times n}(\mathbb{R})\text{:}\) The set of matrices \(E_{ij}\) where \(E_{ij}\) has 1 in the \((i,j)\) position and 0 elsewhere forms a basis. This space has dimension \(mn\text{.}\)
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Note: Unlike the previous examples, \(C(-\infty, \infty)\) and \(C[a,b]\) are infinite-dimensional vector spaces. They do not have finite bases, which makes them more complex to analyze than finite-dimensional spaces.
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To determine whether a set of vectors in an abstract vector space is linearly independent, we set up the equation \(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}\) and determine whether the only solution is the trivial solution. This often reduces to solving a system of linear equations.
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Remark 4.3.20.

Strategy for Testing Linear Independence in Abstract Spaces:
  1. Set up the equation \(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}\text{.}\)
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  2. Use the definition of equality in the vector space to create equations involving \(c_1, c_2, \ldots, c_k\text{.}\)
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  3. For polynomials: equate coefficients of like powers.
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  4. For matrices: equate corresponding entries.
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  5. For functions: use various values of the independent variable or derivatives.
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  6. Solve the resulting system of linear equations.
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  7. If the only solution is trivial, the set is linearly independent; otherwise, it is linearly dependent.
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This theorem justifies using "the dimension" rather than "a dimension"—the dimension of a vector space is uniquely determined by the space itself, not by our choice of basis.
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Definition 4.3.21. Finite-Dimensional and Infinite-Dimensional Vector Spaces.

  • A vector space \(V\) is called finite-dimensional if it has a finite basis, that is, if there exists a basis \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n\}\) with \(n\) vectors, where \(n\) is a non-negative integer. In this case, \(\dim(V) = n\text{.}\)
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  • A vector space \(V\) is called infinite-dimensional if it does not have a finite basis. In this case, we write \(\dim(V) = \infty\text{.}\)
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  • By convention, the zero vector space \(\{\mathbf{0}\}\) has dimension 0: \(\dim(\{\mathbf{0}\}) = 0\text{.}\)
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Remark 4.3.22.

Finite-Dimensional Examples:
  • \(\mathbb{R}^n\) has dimension \(n\text{.}\) The standard basis \(\{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\}\) has \(n\) vectors.
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  • \(\mathcal{P}_n(x)\) has dimension \(n+1\text{.}\) The standard basis \(\{1, x, x^2, \ldots, x^n\}\) has \(n+1\) vectors.
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  • \(M_{m \times n}(\mathbb{R})\) has dimension \(mn\text{.}\) The standard basis \(\{E_{ij} \mid 1 \leq i \leq m, 1 \leq j \leq n\}\) has \(mn\) matrices.
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Infinite-Dimensional Examples:
  • \(C(-\infty, \infty)\text{,}\) the space of all continuous functions on \(\mathbb{R}\text{,}\) is infinite-dimensional. Any finite set of continuous functions cannot span all continuous functions. For example, the set \(\{\sin(x), \cos(x), \sin(2x), \cos(2x), \ldots\}\) is an infinite linearly independent set.
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  • \(C[a,b]\text{,}\) the space of continuous functions on a closed interval, is also infinite-dimensional for the same reason.
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  • \(\mathcal{P}(x)\text{,}\) the space of all polynomials (with no degree restriction), is infinite-dimensional. The infinite set \(\{1, x, x^2, x^3, \ldots\}\) is linearly independent, and no finite subset can span all polynomials.
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Insight 4.3.23. Why Dimension Matters.

The dimension of a vector space tells us:
  • Size of the Space: A space of dimension \(n\) requires exactly \(n\) coordinates to specify any vector (once we choose a basis).
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  • Degrees of Freedom: In applications, the dimension often represents the number of independent parameters or degrees of freedom in a system.
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  • Computational Feasibility: Finite-dimensional spaces allow for computational methods using matrices and linear systems. Infinite-dimensional spaces require more advanced techniques from functional analysis.
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  • Isomorphism: Any \(n\)-dimensional vector space is essentially "the same" as \(\mathbb{R}^n\) (they are isomorphic), meaning all finite-dimensional vector spaces can be studied through the lens of \(\mathbb{R}^n\text{.}\)
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Subsection 4.3.4 Subspaces of Abstract Vector Spaces

Just as we studied subspaces of \(\mathbb{R}^n\text{,}\) we can also identify and work with subspaces of abstract vector spaces. The definition and properties remain the same, but the context changes from column vectors to polynomials, matrices, or functions.
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Definition 4.3.24. Subspace of an Abstract Vector Space.

Let \(V\) be a vector space. A subset \(W\) of \(V\) is called a subspace of \(V\) if \(W\) is itself a vector space under the same operations as \(V\text{.}\)
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Equivalently, \(W\) is a subspace of \(V\) if and only if:
  1. \(\mathbf{0} \in W\) (contains the zero vector)
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  2. If \(\mathbf{u}, \mathbf{v} \in W\text{,}\) then \(\mathbf{u} + \mathbf{v} \in W\) (closed under addition)
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  3. If \(\mathbf{u} \in W\) and \(c \in \mathbb{R}\text{,}\) then \(c\mathbf{u} \in W\) (closed under scalar multiplication)
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Example 4.3.25. Subspaces of \(\mathcal{P}_n(x)\).

Consider the following subsets of \(\mathcal{P}_3(x)\text{:}\)
  1. \(W_1 = \{p(x) \in \mathcal{P}_3(x) \mid p(0) = 0\}\) (polynomials with constant term zero)
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  2. \(W_2 = \{p(x) \in \mathcal{P}_3(x) \mid p(1) = 0\}\) (polynomials that vanish at \(x=1\))
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  3. \(W_3 = \{p(x) \in \mathcal{P}_3(x) \mid p(x) = a_1x + a_3x^3\}\) (polynomials with only odd-degree terms)
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Verify that each of these is a subspace of \(\mathcal{P}_3(x)\text{.}\)
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Solution.
For \(W_1\text{:}\)
  1. The zero polynomial \(p(x) = 0\) satisfies \(p(0) = 0\text{,}\) so \(\mathbf{0} \in W_1\) ✓
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  2. If \(p(x), q(x) \in W_1\text{,}\) then \(p(0) = 0\) and \(q(0) = 0\text{.}\) Thus \((p+q)(0) = p(0) + q(0) = 0 + 0 = 0\text{,}\) so \(p+q \in W_1\) ✓
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  3. If \(p(x) \in W_1\) and \(c \in \mathbb{R}\text{,}\) then \((cp)(0) = c \cdot p(0) = c \cdot 0 = 0\text{,}\) so \(cp \in W_1\) ✓
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For \(W_2\text{:}\)
  1. \(p(x) = 0\) satisfies \(p(1) = 0\text{,}\) so \(\mathbf{0} \in W_2\) ✓
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  2. If \(p(1) = 0\) and \(q(1) = 0\text{,}\) then \((p+q)(1) = p(1) + q(1) = 0 + 0 = 0\) ✓
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  3. If \(p(1) = 0\) and \(c \in \mathbb{R}\text{,}\) then \((cp)(1) = c \cdot p(1) = c \cdot 0 = 0\) ✓
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For \(W_3\text{:}\)
  1. \(p(x) = 0\) can be written as \(0 \cdot x + 0 \cdot x^3\text{,}\) so \(\mathbf{0} \in W_3\) ✓
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  2. If \(p(x) = a_1x + a_3x^3\) and \(q(x) = b_1x + b_3x^3\text{,}\) then \((p+q)(x) = (a_1+b_1)x + (a_3+b_3)x^3 \in W_3\) ✓
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  3. If \(p(x) = a_1x + a_3x^3\) and \(c \in \mathbb{R}\text{,}\) then \(cp(x) = (ca_1)x + (ca_3)x^3 \in W_3\) ✓
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Example 4.3.26. Subspaces of \(M_{2 \times 2}(\mathbb{R})\).

Let \(W\) be the set of all \(2 \times 2\) symmetric matrices:
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\begin{equation*} W = \left\{\begin{bmatrix}a & b\\b & c\end{bmatrix} \mid a, b, c \in \mathbb{R}\right\}. \end{equation*}
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Show that \(W\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\text{.}\)
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Solution.
  1. The zero matrix \(\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\) is symmetric, so \(\mathbf{0} \in W\) ✓
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  2. If \(A = \begin{bmatrix}a_1 & b_1\\b_1 & c_1\end{bmatrix}\) and \(B = \begin{bmatrix}a_2 & b_2\\b_2 & c_2\end{bmatrix}\) are symmetric, then
    \begin{equation*} A + B = \begin{bmatrix}a_1+a_2 & b_1+b_2\\b_1+b_2 & c_1+c_2\end{bmatrix} \end{equation*}
    is also symmetric, so \(A + B \in W\) ✓
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  3. If \(A = \begin{bmatrix}a & b\\b & c\end{bmatrix}\) is symmetric and \(k \in \mathbb{R}\text{,}\) then
    \begin{equation*} kA = \begin{bmatrix}ka & kb\\kb & kc\end{bmatrix} \end{equation*}
    is also symmetric, so \(kA \in W\) ✓
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Example 4.3.27. Finding a Basis for a Polynomial Subspace.

Find a basis for the subspace \(W_3 = \{p(x) \in \mathcal{P}_3(x) \mid p(x) = a_1x + a_3x^3\}\) from the previous example.
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Solution.
Any polynomial in \(W_3\) can be written as:
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\begin{equation*} p(x) = a_1x + a_3x^3 = a_1 \cdot x + a_3 \cdot x^3 \end{equation*}
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This shows that \(W_3 = \operatorname{span}\{x, x^3\}\text{.}\)
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To verify that \(\{x, x^3\}\) is linearly independent, suppose:
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\begin{equation*} c_1 \cdot x + c_2 \cdot x^3 = 0 \end{equation*}
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This means the polynomial \(c_1x + c_2x^3\) is the zero polynomial, which requires \(c_1 = 0\) and \(c_2 = 0\text{.}\)
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Therefore, \(\mathcal{B} = \{x, x^3\}\) is a basis for \(W_3\text{,}\) and \(\dim(W_3) = 2\text{.}\)
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Activity 4.3.1.

(a)
Show that the set \(S\) of symmetric \(2 \times 2\) matrices forms a subspace of \(M_{2 \times 2}(\mathbb{R})\text{.}\)
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(b)
Find a basis for the subspace \(S\text{,}\) and determine its dimension.
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Solution.
The set of symmetric \(2 \times 2\) matrices is a subspace because it contains the zero matrix, is closed under addition, and is closed under scalar multiplication (as shown in the previous example).
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Activity 4.3.2. Finding Bases for Polynomial Subspaces.

Consider the subspace \(W = \{p(x) \in \mathcal{P}_3(x) \mid p(1) = 0\}\text{.}\)
  1. Find a basis for \(W\text{.}\)
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  2. What is the dimension of \(W\text{?}\)
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  3. Express \(p(x) = 2x - 2x^2\) as a linear combination of your basis vectors.
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Hint.
A polynomial \(p(x) = a_0 + a_1x + a_2x^2 + a_3x^3\) satisfies \(p(1) = 0\) if and only if \(a_0 + a_1 + a_2 + a_3 = 0\text{,}\) which means \(a_0 = -a_1 - a_2 - a_3\text{.}\)
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General Strategy for Finding Bases:
  1. Express a general element of the subspace in terms of parameters (free variables)
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  2. Factor out the parameters to identify spanning vectors
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  3. Verify that these spanning vectors are linearly independent
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  4. The resulting set is a basis for the subspace
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This strategy applies universally, whether we’re working with polynomials, matrices, functions, or any other vector space. The key is to understand the structure of the subspace and express it systematically.
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Insight 4.3.28. Why Finding Bases Matters.

Finding a basis for a subspace is fundamental because:
  • Dimension: The size of a basis tells us the dimension, which characterizes the "size" of the subspace
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  • Representation: Bases provide a coordinate system, allowing us to represent abstract vectors as numerical coordinates
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  • Computation: Once we have a basis, we can transform abstract problems into concrete matrix computations
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  • Classification: Subspaces of the same dimension are "essentially the same" (isomorphic)
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  • Applications: In differential equations, the solution space is a subspace, and finding a basis means finding all possible solutions
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