Vector Spaces
1. Vector Space and Subspace
Introduction
A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars while satisfying specific axioms. Examples include Euclidean space \( \mathbb{R}^n \), spaces of functions, and polynomial spaces.
A subspace is a subset of a vector space that is also a vector space under the same operations.
Why are Vector Spaces Important?
- Form the foundation of linear algebra.
- Used in diverse fields such as physics, computer graphics, and data science.
Key Concepts
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Vector Space Axioms:
- Closure under addition and scalar multiplication.
- Associativity and commutativity of addition.
- Existence of additive identity and additive inverses.
- Compatibility of scalar multiplication.
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Subspaces:
- Must contain the zero vector.
- Closed under vector addition and scalar multiplication.
Example
Let \( V = \mathbb{R}^3 \) and consider \( W = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\} \). Prove \( W \) is a subspace.
Solution:
- Zero Vector: \( (0, 0, 0) \in W \).
-
Closure under Addition: If \( (x_1, y_1, z_1), (x_2, y_2, z_2) \in W \), then:
\[ (x_1 + x_2) + (y_1 + y_2) + (z_1 + z_2) = 0 \] -
Closure under Scalar Multiplication: For \( k \in \mathbb{R} \): [ kx + ky + kz = k(x + y + z) = 0 ]
Thus, \( W \) is a subspace.
2. Null Space and Column Space, and Linear Transformations
Null Space and Column Space
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Null Space (Kernel): The set of all solutions to \( A\mathbf{x} = \mathbf{0} \).
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Column Space (Range): The span of the columns of matrix \( A \).
Example
Find the null space and column space of:
Solution:
- Null Space: Solve \( A\mathbf{x} = \mathbf{0} \):
Basis: \( \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} \).
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Column Space: Rank of \( A = 2 \). Basis vectors:
\[ \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} \]
Linear Transformations
A function \( T: V \to W \) is linear if:
- \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
- \( T(c\mathbf{u}) = cT(\mathbf{u}) \).
Example
Show \( T(x, y) = (2x, 3y) \) is a linear transformation.
Solution:
- Addition: \( T((x_1, y_1) + (x_2, y_2)) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2), 3(y_1 + y_2)) \).
- Scalar Multiplication: \( T(c(x, y)) = T(cx, cy) = (2cx, 3cy) \).
3. Linearly Independent Sets, Bases
Linearly Independent Sets
A set of vectors is linearly independent if no vector can be written as a linear combination of the others.
Example
Determine if \( \{(1, 0), (0, 1), (1, 1)\} \) is linearly independent.
Solution: Solve:
This gives \( c_1 + c_3 = 0 \) and \( c_2 + c_3 = 0 \). Only solution: \( c_1 = c_2 = c_3 = 0 \). Independent.
Bases
A basis of a vector space is a linearly independent set that spans the space.
Example
Find a basis for \( \mathbb{R}^2 \).
Solution: Standard basis: \( \{(1, 0), (0, 1)\} \).
4. Coordinate System
Introduction
Coordinates represent a vector relative to a basis.
Example
Given basis \( \{(1, 2), (3, 4)\} \), find coordinates of \( \mathbf{v} = (7, 10) \).
Solution:
Solve \( c_1(1, 2) + c_2(3, 4) = (7, 10) \): \( c_1 = 2, \, c_2 = 1 \)
Coordinates: \( (2, 1) \).
Exercises
Exercise 1
Prove \( W = \{(x, y, z) : x + 2y - z = 0\} \) is a subspace of \( \mathbb{R}^3 \).
Solution:
- Zero Vector: \( (0, 0, 0) \in W \).
- Addition: Closure under addition holds.
- Scalar Multiplication: Closure under scalar multiplication holds.
Exercise 2
Find the null space of \( A = \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} \).
Solution:
Solve \( A\mathbf{x} = \mathbf{0} \):
Basis: \( \begin{bmatrix} 1 \\ -\frac{1}{2} \end{bmatrix} \).
Exercise 3
Find the basis for the column space of:
Solution:
Rank: 2.
Basis vectors: