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Vector Spaces and Linear Transformation

Vector Spaces and Linear Transformations

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.

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