Matrices Applications

Linear Simultaneous Equations

A system of linear equations can be written as a matrix multiplication equation. For example:

\begin{split} 3x + 4y - z = 5 \\ 2x + y - 2z = -12 \\ x + 3z = 4 \\ \pmatrix{3 & 4 & -1 \\ 2 & 1 & -2 \\ 1 & 0 & 3}\pmatrix{x\\y\\z} = \pmatrix{5\\-12\\4} \end{split}

The matrix equation can then be solved.

In general, a system of linear equations can be represented by the matrix equation:

$$MX=B$$

Where M is a matrix of coefficients, X is a column vector of variables, and B is the right-hand side of each equation. This matrix equation can then be solved for the variables:

$$X=M^{-1}B$$

If M is singular, then there is no unique solution to the system.

Linear Transformations

The first column of a transformation matrix is the image of $\left(\begin{array}{c}1\\ 0\end{array}\right)$, and the second column is the image of $\left(\begin{array}{c}0\\ 1\end{array}\right)$.

2D Transformations

Rotations

For a rotation with angle $\theta$ anticlockwise, the rotation matrix is given by: \pmatrix{\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta}

Reflections

A reflection in the $x$ axis is given by: \pmatrix{1 & 0 \\ 0 & -1} A reflection in the $y$ axis is given by: \pmatrix{-1 & 0 \\ 0 & 1} A reflection in the line $y=x$ is given by: \pmatrix{0 & 1 \\ 1 & 0} A reflection in the line $y=-x$ is given by: \pmatrix{0 & -1 \\ -1 & 0}

Enlargement

For an enlargement with scale factor $k$ centred on the origin, the transformation matrix is given by: \pmatrix{k & 0 \\ 0 & k}

Stretches

For a stretch with scale factor $k$ and $x$-axis invariant: \pmatrix{1 & 0 \\ 0 & k} For a stretch with scale factor $k$ and $y$-axis invariant: \pmatrix{k & 0 \\ 0 & 1}

Shears

Shears preserve orientation and area. For a shear with $(0,1)\mapsto (k,1)$ and $x$-axis invariant: \pmatrix{1 & k \\ 0 & 1} For a shear with $(1,0)\mapsto (1,k)$ and $y$-axis invariant: \pmatrix{1 & 0 \\ k & 1}

3D Transformations

Reflections

There are three planes of reflection (needed for this course): the $xy$ plane, the $xz$ plane, and the $yz$ plane. Consider a reflection in the $xy$ plane. This is defined by all points $(x,y,z)$ where $z=0$, so a reflection in the $xy$ plane has the matrix: \pmatrix{1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1} Similar reasoning can be used to find other reflection matrices in 3D.

Rotations

There are three axes of rotation: about the three coordinate axis. The axis of rotation is not affected, so one of the matrix columns is a unit vector. The other matrix elements are populated by the standard 2D rotation matrix $\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$. For example, the matrix for a rotation by $\theta$ anticlockwise about the $x$ axis is:

\pmatrix{1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta}

Note that for rotations about the $y$ axis, the signs on the $\sin \theta$ and $-\sin \theta$ elements are swapped.

Determinant

For a matrix transformation in two dimensions:

• The determinant gives the area scale factor.
• If the determinant is negative then the transformation reverses orientation.

For a matrix transformation in three dimensions:

• The determinant gives the volume scale factor.
• If the determinant is negative then the transformation changes orientation.

For common 2D transformations:

• The determinant of a rotation is 1.
• The determinant of a reflection is -1.
• The determinant of an enlargement with scale factor $k$ is $k^2$.
• The determinant of a stretch with scale factor $k$ is $k$.
• The determinant of a shear is 1.

Combining and Inverses

A matrix M representing transformation A followed by transformation B is given by: $M=BA$ The matrix that is applied first is on the right. This stems from function notation (where the rightmost function in a composite function is applied first).

The inverse transformation is represented by the inverse matrix. For a combined transformation $M=BA$: $M^{-1}=A^{-1}{B}^{-1}$

Invariant Points and Lines

An invariant point of a linear transformation maps to itself. The origin is an invariant point for all linear transformations. An invariant point of matrix M is defined as:

$$M\left(\begin{array}{c}u\\ v\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$$

A line of invariant points is formed when there are infinitely many solutions of the form $v=ku$. Lines of invariant points can be found by applying the transformation to a general point $M\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}x\\ y\end{array}\right)$, and solving simultaneously.

An invariant line is a line where the image of any point on the line is also on the line. Lines of invariant points are a subset of invariant lines. Invariant lines can be found by considering the image of a general line $y=mx$ (or potentially $y=mx+c$).