A vector space is a set of vectors that satisfies specific properties under vector addition and scalar multiplication.
Definition of a Vector Space
A set \( V \) is called a vector space over a field \( \mathbb{R} \) (real numbers) if it satisfies the following properties:
- Closure under addition: If \( \mathbf{u}, \mathbf{v} \in V \), then \( \mathbf{u} + \mathbf{v} \in V \).
- Closure under scalar multiplication: If \( \mathbf{v} \in V \) and \( c \in \mathbb{R} \), then \( c\mathbf{v} \in V \).
- Associativity: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
- Commutativity: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
- Existence of a zero vector: There exists a vector \( \mathbf{0} \) such that \( \mathbf{v} + \mathbf{0} = \mathbf{v} \).
- Existence of additive inverses: For each \( \mathbf{v} \), there exists \( -\mathbf{v} \) such that \( \mathbf{v} + (-\mathbf{v}) = \mathbf{0} \).
- Distributivity: \( c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} \).
Example of a Vector Space
The set of all 2D vectors \( \mathbb{R}^2 \), defined as:
\[ V = \left\{ \begin{bmatrix} x \\ y \end{bmatrix} \mid x, y \in \mathbb{R} \right\} \]is a vector space because it satisfies all vector space properties.
2. Linear Transformations
A linear transformation is a function \( T: V \to W \) that preserves vector addition and scalar multiplication.
Definition of a Linear Transformation
A function \( T: V \to W \) is a linear transformation if for all \( \mathbf{u}, \mathbf{v} \in V \) and \( c \in \mathbb{R} \), it satisfies:
\[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \] \[ T(c\mathbf{v}) = cT(\mathbf{v}) \]Example of a Linear Transformation
The transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by:
\[ T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x + y \\ 3x - y \end{bmatrix} \]is linear because it satisfies both properties.
Practice Problems
Problem 1: Determine if the set of all polynomials of degree at most 2, denoted as:
\[ V = \{ a + bx + cx^2 \mid a, b, c \in \mathbb{R} \} \]forms a vector space under usual polynomial addition and scalar multiplication.
Solution:
Yes, this set satisfies all vector space properties:
- It is closed under addition and scalar multiplication.
- It contains a zero polynomial \( 0 + 0x + 0x^2 \).
- Every polynomial has an additive inverse.
Thus, it forms a vector space.
Problem 2: Show that the transformation:
\[ T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x + y \\ 2x - y \end{bmatrix} \]is a linear transformation.
Solution:
Check the two properties:
- It satisfies \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
- It satisfies \( T(c\mathbf{v}) = cT(\mathbf{v}) \).
Thus, \( T \) is a linear transformation.