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}

Tudo estava claro?

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Obrigado pelo seu feedback!

Seção 4. Capítulo 5

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Pergunte o que quiser ou experimente uma das perguntas sugeridas para iniciar nosso bate-papo

Suggested prompts:

Can you explain how to multiply a matrix by a vector step by step?

What are some other common matrix transformations besides scaling and rotation?

How do you interpret the result of a matrix transformation geometrically?

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Introduction to Matrix Transformations

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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}

Tudo estava claro?

Como podemos melhorá-lo?

Obrigado pelo seu feedback!

Seção 4. Capítulo 5

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?