Introduction to Matrix Transformations

Matrices allow us to represent and manipulate linear transformations in space. They are core to systems of equations, geometry, and data science workflows.

What Is a Matrix?

A matrix is a rectangular array of numbers used to represent transformations or systems of 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:

$s_x$ is the scale factor in the x-direction;
$s_y$ is 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}

Quiz

What is the matrix form of this system of equations?

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

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

B) …

C) …

D) …

Correct Answer: A

Correct Answer: B

1. Given the transformation matrix:

M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

What is the result of the transformation?

2. What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

3. If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

4. Which matrix correctly rotates a vector $90°$ counterclockwise?

5. What is the matrix form of this system of equations?

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

Given the transformation matrix:

M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

What is the result of the transformation?

Select the correct answer

M \vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 8 \\ 2 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 4 \\ 8 \end{bmatrix}

What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

Select the correct answer

\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}

\begin{bmatrix} 3 & 0 \\ 0 & 0.5 \end{bmatrix}

\begin{bmatrix}0 & 0.5 \\ 3 & 0 \end{bmatrix}

\begin{bmatrix} 3 & 0.5 \\ 0 & 0.5 \end{bmatrix}

If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

Select the correct answer

(2, 3)

(4, 6)

(6, 9)

(1, 1)

Which matrix correctly rotates a vector $90°$ counterclockwise?

Select the correct answer

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

\begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix}

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

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

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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Introduction to Matrix Transformations

Pyyhkäise näyttääksesi valikon

Matrices allow us to represent and manipulate linear transformations in space. They are core to systems of equations, geometry, and data science workflows.

What Is a Matrix?

A matrix is a rectangular array of numbers used to represent transformations or systems of 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:

$s_x$ is the scale factor in the x-direction;
$s_y$ is 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}

Quiz

What is the matrix form of this system of equations?

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

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

B) …

C) …

D) …

Correct Answer: A

Correct Answer: B

1. Given the transformation matrix:

M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

What is the result of the transformation?

2. What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

3. If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

4. Which matrix correctly rotates a vector $90°$ counterclockwise?

5. What is the matrix form of this system of equations?

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

Given the transformation matrix:

M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

What is the result of the transformation?

Select the correct answer

M \vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 8 \\ 2 \end{bmatrix}

M \vec{v} = \begin{bmatrix} 4 \\ 8 \end{bmatrix}

What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

Select the correct answer

\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}

\begin{bmatrix} 3 & 0 \\ 0 & 0.5 \end{bmatrix}

\begin{bmatrix}0 & 0.5 \\ 3 & 0 \end{bmatrix}

\begin{bmatrix} 3 & 0.5 \\ 0 & 0.5 \end{bmatrix}

If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

Select the correct answer

(2, 3)

(4, 6)

(6, 9)

(1, 1)

Which matrix correctly rotates a vector $90°$ counterclockwise?

Select the correct answer

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

\begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix}

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

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

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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Introduction to Matrix Transformations

What Is a Matrix?

Matrix Representation of Linear Systems

Matrix Multiplication Breakdown

Example System in Matrix Form

Matrices as Transformations

Scaling with Matrices

Rotation with Matrices

Reflection over the x-axis

Shearing Transformation (x-direction shear)

Identity Transformation

Quiz

1. Given the transformation matrix:

2. What matrix represents the transformation that scales by 333 in xxx and 12\frac{1}{2}21​ in yyy?

3. If you scale vector (2,3)(2, 3)(2,3) by a factor of 222 in both xxx and yyy directions, what is the resulting vector?

4. Which matrix correctly rotates a vector 90°90°90° counterclockwise?

5. What is the matrix form of this system of equations?

Awesome!

Introduction to Matrix Transformations

What Is a Matrix?

Matrix Representation of Linear Systems

Matrix Multiplication Breakdown

Example System in Matrix Form

Matrices as Transformations

Scaling with Matrices

Rotation with Matrices

Reflection over the x-axis

Shearing Transformation (x-direction shear)

Identity Transformation

Quiz

1. Given the transformation matrix:

2. What matrix represents the transformation that scales by 333 in xxx and 12\frac{1}{2}21​ in yyy?

3. If you scale vector (2,3)(2, 3)(2,3) by a factor of 222 in both xxx and yyy directions, what is the resulting vector?

4. Which matrix correctly rotates a vector 90°90°90° counterclockwise?

5. What is the matrix form of this system of equations?

2. What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

3. If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

4. Which matrix correctly rotates a vector $90°$ counterclockwise?

2. What matrix represents the transformation that scales by $3$ in $x$ and $\frac{1}{2}$ in $y$ ?

3. If you scale vector $(2, 3)$ by a factor of $2$ in both $x$ and $y$ directions, what is the resulting vector?

4. Which matrix correctly rotates a vector $90°$ counterclockwise?