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bookIntroduction to Matrix Transformations

Matrix Equations

A matrix equation can be written as:

Ax⃗=b⃗A \vec{x} = \vec{b}

Where:

  • AA is the coefficient matrix;
  • xβƒ—\vec{x} is the vector of variables;
  • bβƒ—\vec{b} is the vector of constants.

Matrix Representation of Linear Systems

Consider the linear system:

2x+y=5xβˆ’y=12x + y = 5 \\ x - y = 1

This can be rewritten as:

[211βˆ’1][xy]=[51]\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:

[abcd][xy]=[ax+bycx+dy]=x[ac]+y[bd]\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=74xβˆ’y=53x + 2y = 7 \\ 4x - y = 5

Can be expressed as:

[324βˆ’1][xy]=[75]\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=[abcd],Β Β v1βƒ—=[11],Β Β v2βƒ—=[112]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=[sx00sy]S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}

Where:

  • sxs_x - the scale factor in the x-direction;
  • sys_y - the scale factor in the y-direction.

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

S=[2002],v⃗=[23]S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

Sv⃗=[46]S \vec{v} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}

Rotation with Matrices

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

R=[cosβ‘ΞΈβˆ’sin⁑θsin⁑θcos⁑θ]R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Example: rotate (2, 3) by 90Β°:

R=[cos⁑90ΒΊβˆ’sin⁑90ΒΊsin⁑90ΒΊcos⁑90ΒΊ]=[0βˆ’110],vβƒ—=[23]R = \begin{bmatrix} \cos90ΒΊ & -\sin90ΒΊ \\ \sin90ΒΊ & \cos90ΒΊ \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

Rvβƒ—=[βˆ’32]R \vec{v} = \begin{bmatrix} -3 \\ 2 \end{bmatrix}

Reflection over the x-axis

Reflection matrix:

M=[100βˆ’1],M = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix},

Using v⃗=(2,3)\vec{v} = (2, 3):

Mvβƒ—=[2βˆ’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=[1k01]M = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}

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

Mv⃗=[6.53]M \vec{v} = \begin{bmatrix} 6.5 \\ 3 \end{bmatrix}

Identity Transformation

The identity matrix performs no transformation:

I=[1001]I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

For any vector v⃗\vec{v}:

Iv⃗=v⃗I \vec{v} = \vec{v}
question mark

What is the matrix form of this system of equations?

2x+y=5xβˆ’y=12x + y = 5 \\ x - y = 1

Select the correct answer

Everything was clear?

How can we improve it?

Thanks for your feedback!

SectionΒ 4. ChapterΒ 5

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bookIntroduction to Matrix Transformations

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Matrix Equations

A matrix equation can be written as:

Ax⃗=b⃗A \vec{x} = \vec{b}

Where:

  • AA is the coefficient matrix;
  • xβƒ—\vec{x} is the vector of variables;
  • bβƒ—\vec{b} is the vector of constants.

Matrix Representation of Linear Systems

Consider the linear system:

2x+y=5xβˆ’y=12x + y = 5 \\ x - y = 1

This can be rewritten as:

[211βˆ’1][xy]=[51]\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:

[abcd][xy]=[ax+bycx+dy]=x[ac]+y[bd]\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=74xβˆ’y=53x + 2y = 7 \\ 4x - y = 5

Can be expressed as:

[324βˆ’1][xy]=[75]\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=[abcd],Β Β v1βƒ—=[11],Β Β v2βƒ—=[112]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=[sx00sy]S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}

Where:

  • sxs_x - the scale factor in the x-direction;
  • sys_y - the scale factor in the y-direction.

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

S=[2002],v⃗=[23]S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

Sv⃗=[46]S \vec{v} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}

Rotation with Matrices

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

R=[cosβ‘ΞΈβˆ’sin⁑θsin⁑θcos⁑θ]R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Example: rotate (2, 3) by 90Β°:

R=[cos⁑90ΒΊβˆ’sin⁑90ΒΊsin⁑90ΒΊcos⁑90ΒΊ]=[0βˆ’110],vβƒ—=[23]R = \begin{bmatrix} \cos90ΒΊ & -\sin90ΒΊ \\ \sin90ΒΊ & \cos90ΒΊ \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

Then:

Rvβƒ—=[βˆ’32]R \vec{v} = \begin{bmatrix} -3 \\ 2 \end{bmatrix}

Reflection over the x-axis

Reflection matrix:

M=[100βˆ’1],M = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix},

Using v⃗=(2,3)\vec{v} = (2, 3):

Mvβƒ—=[2βˆ’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=[1k01]M = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}

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

Mv⃗=[6.53]M \vec{v} = \begin{bmatrix} 6.5 \\ 3 \end{bmatrix}

Identity Transformation

The identity matrix performs no transformation:

I=[1001]I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

For any vector v⃗\vec{v}:

Iv⃗=v⃗I \vec{v} = \vec{v}
question mark

What is the matrix form of this system of equations?

2x+y=5xβˆ’y=12x + y = 5 \\ x - y = 1

Select the correct answer

Everything was clear?

How can we improve it?

Thanks for your feedback!

SectionΒ 4. ChapterΒ 5
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