Using Matrices as Functions of Vectors

Concept of Matrices as Functions

A matrix can be seen as a function that operates on a vector to produce another vector as output. In this lecture, we explored using matrices to transform vectors.

Matrix Representation

• The relationship between a matrix, an input vector, and an output vector is illustrated as follows:

  • Input Vector: A vector ( $\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}$ )
  • Output Vector: A vector ( $\mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix}$ )

• Matrix multiplication can be represented as: $A \cdot \mathbf{x} = \mathbf{y}$ Where ( $A$ ) is a matrix that transforms ( $\mathbf{x}$ ) into ( $\mathbf{y}$ ).

Example 1: Solving a System of Linear Equations

For the system: $x + 2y = 3 \quad \text{(Equation 1)}$ $2x + 3y = 5 \quad \text{(Equation 2)}$

We can represent the system in matrix form as:

• Interpretation:
  • The matrix on the left represents the coefficients of ( $x$ ) and ( $y$ ) in the system of equations.
  • The vector on the left ( $\begin{bmatrix} x \ y \end{bmatrix}$ ) is the input vector (unknowns).
  • The vector on the right ( $\begin{bmatrix} 3 \ 5 \end{bmatrix}$ ) is the output vector (constant terms from the equations).

The goal is to find ( $x$ ) and ( $y$ ) that satisfy this equation.

Identity Matrix

The identity matrix ( I ) has the following property:

$I \cdot \mathbf{v} = \mathbf{v}$

• Example: ( $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix}$ )
• It does not change the vector when multiplied.

Matrix Operations

• Example 2: Vector Transformation

  A vector can be scaled by multiplying it by a scalar. For example:

  $$\frac{1}{2} \cdot \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \ \frac{1}{2} \end{bmatrix}$$
  • This scales the vector by ( $\frac{1}{2}$ ).

• Example 3: Scalar Multiplication of a Matrix

  Multiplying a matrix by a scalar is another operation:

  $$1/5 \cdot \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 1/5 \ 1/5 \end{bmatrix}$$

• Example 4: Zero Vector

  • When a matrix is multiplied by a zero vector, the result is always a zero vector: $$\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}$$

Graphical Interpretation

• The concept of a matrix as a function of vectors can also be visualized geometrically. A matrix can transform vectors by scaling, rotating, or translating them in space.
• Example: Consider a vector ( $v = \begin{bmatrix} 1 \ 1 \end{bmatrix}$ ), it can be scaled or transformed by a matrix such as: $$A = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix}$$ The transformed vector is: $$A \cdot v = \begin{bmatrix} 2 \ 2 \end{bmatrix}$$ This is a scaling transformation.

Key Concepts

• Matrices as functions: A matrix can be thought of as a function that transforms input vectors into output vectors.
• Identity matrix: The matrix that leaves vectors unchanged when multiplied.
• Scalar multiplication: Matrices and vectors can be scaled by multiplying them by scalars.
• Zero vector: Multiplying any matrix by a zero vector results in a zero vector.