Vector Spaces in Linear Algebra - Examples with Solutions

Vector spaces form the foundational structure of linear algebra, unifying diverse mathematical objects such as real numbers, vectors, matrices, polynomials, and functions under common algebraic laws and algorithms.

Table of Contents

Introduction to Vector Spaces

Consider the following algebraic equations:

Because these distinct mathematical objects obey common laws of addition and scalar multiplication, we can solve them and many other complex problems using unified algorithms. Classifying sets by their algebraic properties streamlines problem-solving across matrices, polynomials, vectors, geometric planes, and functions.

Definition of a Vector Space

In linear algebra, vector spaces are denoted by capital letters (e.g., \( V \)) and their elements (called vectors) by bold lowercase letters (e.g., \( \mathbf{u}, \mathbf{v}, \mathbf{w} \)).

A nonempty set \( V \) equipped with operations of addition (\( + \)) and scalar multiplication (\( \cdot \)) is called a vector space if it satisfies the following conditions:

A) Closure Axioms:

  1. Closure under addition: For any \( \mathbf{u}, \mathbf{v} \in V \), the sum \( \mathbf{u} + \mathbf{v} \) is an element of \( V \).
  2. Closure under scalar multiplication: For any \( \mathbf{u} \in V \) and scalar \( r \in \mathbb{R} \), the product \( r \cdot \mathbf{u} \) is an element of \( V \).

B) Algebraic Axioms: For any \( \mathbf{u}, \mathbf{v}, \mathbf{w} \in V \) and scalars \( r, s \in \mathbb{R} \):

  1. Commutativity of vector addition: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \)
  2. Associativity of vector addition: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \)
  3. Associativity of scalar multiplication: \( r \cdot (s \cdot \mathbf{u}) = (r \cdot s) \cdot \mathbf{u} \)
  4. Existence of a zero vector: There exists a zero vector \( \mathbf{0} \in V \) such that \( \mathbf{u} + \mathbf{0} = \mathbf{u} \) for all \( \mathbf{u} \in V \).
  5. Existence of negative vectors: For each \( \mathbf{u} \in V \), there exists a negative vector \( -\mathbf{u} \in V \) such that \( \mathbf{u} + (-\mathbf{u}) = \mathbf{0} \).
  6. Distributivity over vector addition: \( r \cdot (\mathbf{u} + \mathbf{v}) = r \cdot \mathbf{u} + r \cdot \mathbf{v} \)
  7. Distributivity over scalar addition: \( (r + s) \cdot \mathbf{u} = r \cdot \mathbf{u} + s \cdot \mathbf{u} \)
  8. Scalar identity: \( 1 \cdot \mathbf{u} = \mathbf{u} \)

Note: Elements of a vector space do not have to be geometric vectors of 2 or 3 dimensions; they can be matrices, functions, polynomials, or differential equation solutions.

Key Properties of Vector Spaces

The following properties follow directly from the ten axioms and hold in every vector space:

  1. Uniqueness of the zero vector: There is exactly one zero vector \( \mathbf{0} \) in \( V \).
  2. Uniqueness of the negative: Each vector \( \mathbf{u} \) has exactly one negative \( -\mathbf{u} \).
  3. Zero scalar: \( 0 \cdot \mathbf{u} = \mathbf{0} \) for every vector \( \mathbf{u} \in V \).
  4. Zero vector scalar: \( r \cdot \mathbf{0} = \mathbf{0} \) for every scalar \( r \).
  5. Negative scalar: \( (-1) \cdot \mathbf{u} = -\mathbf{u} \) for every vector \( \mathbf{u} \in V \).
  6. Cancellation: If \( \mathbf{u} + \mathbf{w} = \mathbf{v} + \mathbf{w} \), then \( \mathbf{u} = \mathbf{v} \).

Note: A subspace is a subset of a vector space that is itself a vector space under the inherited operations. To verify a subspace, it suffices to check closure under addition, closure under scalar multiplication, and the presence of the zero vector.

Examples of Vector Spaces

Example 1: Standard Examples of Vector Spaces

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The following mathematical structures are well-known vector spaces:

  1. The set of all real numbers \( \mathbb{R} \) with standard addition and multiplication.
  2. The set of all complex numbers \( \mathbb{C} \).
  3. The set of all polynomials \( P_n \) of degree \( \le n \) with real coefficients.
  4. The set of all \( n \)-dimensional vectors \( \mathbb{R}^n \).
  5. The set of all \( m \times n \) matrices.
  6. The set of all real functions satisfying a homogeneous linear differential equation.

Example 2: Proving that \( 2 \times 2 \) Matrices Form a Vector Space

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Prove that the set \( V \) of all \( 2 \times 2 \) matrices, equipped with standard matrix addition and scalar multiplication, is a vector space.

Solution

Let \( V \) be the set of all \( 2 \times 2 \) matrices of the form \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) where \( a, b, c, d \in \mathbb{R} \).

1) Closure under addition:

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} = \begin{bmatrix} a+a' & b+b' \\ c+c' & d+d' \end{bmatrix} \]

The sum of two \( 2 \times 2 \) matrices is another \( 2 \times 2 \) matrix, so \( V \) is closed under addition.

2) Closure under scalar multiplication:

\[ r \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ra & rb \\ rc & rd \end{bmatrix} \]

The scalar multiple is a \( 2 \times 2 \) matrix, so \( V \) is closed under scalar multiplication.

3) Commutativity: Matrix addition is component-wise addition of real numbers, which is commutative:

\[ \begin{bmatrix} a+a' & b+b' \\ c+c' & d+d' \end{bmatrix} = \begin{bmatrix} a'+a & b'+b \\ c'+c & d'+d \end{bmatrix} = \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} + \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

4) Associativity of addition: Follows directly from the associativity of real number addition.

5) Associativity of scalar multiplication: \( r(sA) = (rs)A \) holds component-wise.

6) Zero vector: The zero matrix \( \mathbf{0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \) acts as the additive identity.

7) Negative vector: For any matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the negative matrix \( -A = \begin{bmatrix} -a & -b \\ -c & -d \end{bmatrix} \) satisfies \( A + (-A) = \mathbf{0} \).

8 & 9) Distributivity laws: Hold component-wise from real number arithmetic.

10) Scalar identity: \( 1 \cdot A = A \).

Thus, \( V \) satisfies all 10 axioms and is a vector space.

Example 3: Continuous Functions Form a Vector Space

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Show that the set of all real-valued functions continuous on \( (-\infty, \infty) \) forms a vector space under function addition and scalar multiplication.

Solution

From calculus, if \( f \) and \( g \) are continuous functions on \( (-\infty, \infty) \) and \( r \) is a real number:

  • \( (f + g)(x) = f(x) + g(x) \) is continuous on \( (-\infty, \infty) \).
  • \( (rf)(x) = r \cdot f(x) \) is continuous on \( (-\infty, \infty) \).

Thus, closure under addition and scalar multiplication holds. The remaining 8 axioms are automatically satisfied by real-valued function arithmetic.

Example 4: Polynomials of Degree \( \le 3 \) Form a Vector Space

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Show that the set of all polynomials with degree \( n \le 3 \) forms a vector space.

Solution

The sum of two polynomials of degree less than or equal to 3 is still a polynomial of degree less than or equal to 3. Similarly, multiplying such a polynomial by a real number maintains a degree less than or equal to 3. The algebraic axioms hold true for polynomial arithmetic, confirming it is a vector space.

Note: The zero polynomial \( P(x) = 0 \) is included in this set and serves as the zero vector.

Example 5: Why Degree 4 Polynomials Do NOT Form a Vector Space

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Show that the set of polynomials of exact degree \( n = 4 \) is NOT a vector space.

Solution

To be a vector space, the set must be closed under addition. Consider these two polynomials of degree 4:

\[ P(x) = -2x^4 + 3x^2 - 2x + 6, \quad Q(x) = 2x^4 - 5x^2 + 10 \]

Adding them together:

\[ P(x) + Q(x) = (-2x^4 + 3x^2 - 2x + 6) + (2x^4 - 5x^2 + 10) = -5x^2 - 2x + 16 \]

The resulting polynomial has degree 2, not 4. Because the set is not closed under addition, it is not a vector space.

Additional issue: The set also lacks a zero vector, since the zero polynomial does not have degree 4.

Example 6: Why the Set of Integers Is Not a Vector Space

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Show that the set of integers is NOT a vector space under standard addition and scalar multiplication.

Solution

A vector space must be closed under scalar multiplication by any real number. Let integer \( x = -2 \) and real scalar \( r = \sqrt{3} \):

\[ r \cdot x = \sqrt{3} \cdot (-2) = -2\sqrt{3} \]

The result is an irrational number, which is not an integer. Therefore, the set of integers is not closed under scalar multiplication and is not a vector space.

Frequently Asked Questions

What is a vector space in linear algebra?

A vector space is a nonempty set \( V \) equipped with addition and scalar multiplication that satisfies ten axioms: two closure axioms and eight algebraic axioms (commutativity, associativity, zero vector, negative vectors, distributivity, and scalar identity).

What are the two closure axioms of a vector space?

The two closure axioms are: (1) Closure under addition: for any \( \mathbf{u}, \mathbf{v} \in V \), \( \mathbf{u} + \mathbf{v} \in V \). (2) Closure under scalar multiplication: for any \( \mathbf{u} \in V \) and scalar \( r \), \( r \cdot \mathbf{u} \in V \).

Why is the set of integers not a vector space?

The set of integers is not a vector space because it is not closed under scalar multiplication by arbitrary real numbers. For example, multiplying the integer \( -2 \) by the real scalar \( \sqrt{3} \) gives \( -2\sqrt{3} \), which is not an integer.

Do polynomials form a vector space?

Yes. The set of all polynomials of degree less than or equal to \( n \) (for fixed \( n \)) forms a vector space because the sum of two such polynomials and any scalar multiple of such a polynomial remain polynomials of degree at most \( n \).

What are some examples of vector spaces?

Common examples of vector spaces include: real numbers \( \mathbb{R} \), complex numbers \( \mathbb{C} \), \( n \)-dimensional vectors \( \mathbb{R}^n \), \( m \times n \) matrices, polynomials of degree at most \( n \), continuous functions on an interval, and solutions to homogeneous linear differential equations.

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