Learn Introduction to Matrix Transformations

Matrix Equations

A matrix equation can be written as:

A \vec{x} = \vec{b}

Where:

$A$ is the coefficient matrix;
$\vec{x}$ is the vector of variables;
$\vec{b}$ is the vector of constants.

Matrix Representation of Linear Systems

Consider the linear system:

2x + y = 5 \\ x - y = 1

This can be rewritten as:

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

Matrix Multiplication Breakdown

The multiplication of a matrix with a vector represents a linear combination:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} = x \begin{bmatrix} a \\ c \end{bmatrix} + y\begin{bmatrix} b \\ d \end{bmatrix}

Example System in Matrix Form

The system:

3x + 2y = 7 \\ 4x - y = 5

Can be expressed as:

\begin{bmatrix} 3 & 2 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 5 \end{bmatrix}

Matrices as Transformations

A matrix transforms vectors in space.

For example:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},\ \ \vec{v_1} = \begin{bmatrix}1 \\ 1\end{bmatrix},\ \ \vec{v_2} = \begin{bmatrix}1 \\ \frac{1}{2}\end{bmatrix}

This matrix defines how the axes transform under multiplication.

Scaling with Matrices

To apply scaling to a vector use:

S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}

Where:

$s_x$ - the scale factor in the x-direction;
$s_y$ - the scale factor in the y-direction.

Example: scaling point (2, 3) by 2:

S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

S \vec{v} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}

Rotation with Matrices

To rotate a vector by angle $\theta$ around the origin:

R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Example: rotate (2, 3) by 90°:

R = \begin{bmatrix} \cos90º & -\sin90º \\ \sin90º & \cos90º \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

R \vec{v} = \begin{bmatrix} -3 \\ 2 \end{bmatrix}

Reflection over the x-axis

Reflection matrix:

M = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix},

Using $\vec{v} = (2, 3)$ :

M \vec{v} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}

Shearing Transformation (x-direction shear)

Shearing shifts one axis based on the other.

To shear in the x-direction:

M = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}

If $k = 1.5$ and $\vec{v} = (2, 3)$ :

M \vec{v} = \begin{bmatrix} 6.5 \\ 3 \end{bmatrix}

Identity Transformation

The identity matrix performs no transformation:

I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

For any vector $\vec{v}$ :

I \vec{v} = \vec{v}

What is the matrix form of this system of equations?

2x + y = 5 \\ x - y = 1

Select the correct answer

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

\begin{bmatrix} 5 & 1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

\begin{bmatrix} 5 & 2 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}

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Thanks for your feedback!

Section 4. Chapter 5

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Matrix Equations

A matrix equation can be written as:

A \vec{x} = \vec{b}

Where:

$A$ is the coefficient matrix;
$\vec{x}$ is the vector of variables;
$\vec{b}$ is the vector of constants.

Matrix Representation of Linear Systems

Consider the linear system:

2x + y = 5 \\ x - y = 1

This can be rewritten as:

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

Matrix Multiplication Breakdown

The multiplication of a matrix with a vector represents a linear combination:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} = x \begin{bmatrix} a \\ c \end{bmatrix} + y\begin{bmatrix} b \\ d \end{bmatrix}

Example System in Matrix Form

The system:

3x + 2y = 7 \\ 4x - y = 5

Can be expressed as:

\begin{bmatrix} 3 & 2 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 5 \end{bmatrix}

Matrices as Transformations

A matrix transforms vectors in space.

For example:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},\ \ \vec{v_1} = \begin{bmatrix}1 \\ 1\end{bmatrix},\ \ \vec{v_2} = \begin{bmatrix}1 \\ \frac{1}{2}\end{bmatrix}

This matrix defines how the axes transform under multiplication.

Scaling with Matrices

To apply scaling to a vector use:

S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}

Where:

$s_x$ - the scale factor in the x-direction;
$s_y$ - the scale factor in the y-direction.

Example: scaling point (2, 3) by 2:

S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

S \vec{v} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}

Rotation with Matrices

To rotate a vector by angle $\theta$ around the origin:

R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Example: rotate (2, 3) by 90°:

R = \begin{bmatrix} \cos90º & -\sin90º \\ \sin90º & \cos90º \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

R \vec{v} = \begin{bmatrix} -3 \\ 2 \end{bmatrix}

Reflection over the x-axis

Reflection matrix:

M = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix},

Using $\vec{v} = (2, 3)$ :

M \vec{v} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}

Shearing Transformation (x-direction shear)

Shearing shifts one axis based on the other.

To shear in the x-direction:

M = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}

If $k = 1.5$ and $\vec{v} = (2, 3)$ :

M \vec{v} = \begin{bmatrix} 6.5 \\ 3 \end{bmatrix}

Identity Transformation

The identity matrix performs no transformation:

I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

For any vector $\vec{v}$ :

I \vec{v} = \vec{v}

What is the matrix form of this system of equations?

2x + y = 5 \\ x - y = 1

Select the correct answer

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

\begin{bmatrix} 5 & 1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

\begin{bmatrix} 5 & 2 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}

\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}

Everything was clear?

Thanks for your feedback!

Section 4. Chapter 5