Tutorial on Vector Spaces in Linear Algebra | Generated by AI
This tutorial covers key concepts related to vector spaces, including subspaces, basis, dimension, and linear independence. These topics are fundamental in linear algebra and are essential for understanding more advanced concepts.
1. Vector Spaces
A vector space (or linear space) is a set of vectors that satisfies certain properties under vector addition and scalar multiplication.
Definition
A vector space \( V \) over a field \( \mathbb{F} \) (e.g., real numbers \( \mathbb{R} \) or complex numbers \( \mathbb{C} \)) is a set of elements (vectors) along with two operations:
- Vector Addition: \( \mathbf{u} + \mathbf{v} \) for \( \mathbf{u}, \mathbf{v} \in V \).
- Scalar Multiplication: \( c \mathbf{v} \) for \( c \in \mathbb{F} \) and \( \mathbf{v} \in V \).
These operations must satisfy the following axioms:
- Associativity of Addition: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
- Commutativity of Addition: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
- Existence of Zero Vector: There exists a vector \( \mathbf{0} \) such that \( \mathbf{v} + \mathbf{0} = \mathbf{v} \) for all \( \mathbf{v} \).
- Existence of Additive Inverses: For every \( \mathbf{v} \), there exists \( -\mathbf{v} \) such that \( \mathbf{v} + (-\mathbf{v}) = \mathbf{0} \).
- Distributivity of Scalar Multiplication over Vector Addition: \( c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} \).
- Distributivity of Scalar Multiplication over Field Addition: \( (a + b) \mathbf{v} = a\mathbf{v} + b\mathbf{v} \).
- Associativity of Scalar Multiplication: \( a(b\mathbf{v}) = (ab)\mathbf{v} \).
- Multiplicative Identity: \( 1 \mathbf{v} = \mathbf{v} \).
Examples of Vector Spaces
- \( \mathbb{R}^n \) (n-dimensional Euclidean space)
- The space of polynomials of degree \( \leq n \).
- The set of \( m \times n \) matrices.
- The set of continuous functions.
2. Subspaces
A subspace is a subset \( W \) of a vector space \( V \) that is itself a vector space under the same operations.
Subspace Conditions
A non-empty subset \( W \) of \( V \) is a subspace if:
- Closed under addition: If \( \mathbf{u}, \mathbf{v} \in W \), then \( \mathbf{u} + \mathbf{v} \in W \).
- Closed under scalar multiplication: If \( \mathbf{v} \in W \) and \( c \in \mathbb{F} \), then \( c\mathbf{v} \in W \).
Examples of Subspaces
- The set of all vectors in \( \mathbb{R}^3 \) of the form \( (x, 0, 0) \).
- The set of all polynomials with only even-degree terms.
- The set of solutions to a homogeneous linear equation.
3. Linear Independence
A set of vectors \( { \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k } \) in \( V \) is linearly dependent if there exist scalars \( c_1, c_2, \dots, c_k \), not all zero, such that:
[ c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = 0 ]
If the only solution to this equation is \( c_1 = c_2 = \dots = c_k = 0 \), the vectors are linearly independent.
Examples
- The vectors \( (1,0) \) and \( (0,1) \) in \( \mathbb{R}^2 \) are linearly independent.
- The vectors \( (1,1) \), \( (2,2) \) in \( \mathbb{R}^2 \) are linearly dependent because \( 2(1,1) - (2,2) = (0,0) \).
4. Basis of a Vector Space
A basis of a vector space \( V \) is a set of linearly independent vectors that span \( V \). This means:
- The basis vectors are linearly independent.
- Every vector in \( V \) can be expressed as a linear combination of the basis vectors.
Examples
- The standard basis for \( \mathbb{R}^2 \) is \( { (1,0), (0,1) } \).
- The standard basis for \( \mathbb{R}^3 \) is \( { (1,0,0), (0,1,0), (0,0,1) } \).
5. Dimension of a Vector Space
The dimension of a vector space \( V \), denoted \( \dim(V) \), is the number of vectors in any basis for \( V \).
Examples
- \( \dim(\mathbb{R}^n) = n \).
- The space of polynomials of degree \( \leq 2 \) has dimension 3, with basis \( {1, x, x^2} \).
- The set of solutions to a homogeneous system of 3 equations in 5 unknowns forms a subspace of dimension 2.
Summary of Key Points
| Concept | Definition |
|---|---|
| Vector Space | A set of vectors closed under addition and scalar multiplication. |
| Subspace | A subset of a vector space that is itself a vector space. |
| Linear Independence | A set of vectors is independent if no vector can be written as a linear combination of the others. |
| Basis | A minimal set of vectors that spans the vector space. |
| Dimension | The number of vectors in a basis of the space. |
Practice Problems
- Determine whether the set of vectors \( {(1,2,3), (4,5,6), (7,8,9)} \) in \( \mathbb{R}^3 \) is linearly independent.
- Find a basis for the subspace of \( \mathbb{R}^3 \) spanned by \( {(1,2,3), (2,4,6)} \).
- Find the dimension of the space of solutions to the system: [ x + y + z = 0 ] [ 2x + 3y + 5z = 0 ]
- Verify whether the set \( {1, x, x^2, x^3} \) forms a basis for the space of polynomials of degree \( \leq 3 \).
Let me know if you need explanations for any of these problems! 🚀