Linear Algebra: Solving Linear Equations - Vectors and Linear Equations #1

Concepts

Multiplication by rows

\(Ax\) comes from dot products, each row times the column \(x\):

\[Ax = \begin{pmatrix}(row 1) \cdot x \\ (row 2) \cdot x \\ (row 3) \cdot x \end{pmatrix}\]

Multiplication by columns

\(Ax\) is a combination of column vectors:

\[Ax = x(\text{column 1}) + y(\text{column 2}) + z(\text{column 3})\]

Questions

Question 1

Write \(2x+3y+z+5t=8\) as a matrix \(A\) (how many rows?) multiplying the column vector \(x = (x, y, z, t)\) to produce \(b\). The solutions \(x\) fill a plane or “hyperplane” in 4-dimensional space. The plane is 3-dimensional with no 4D volume.

Question 2

Find the matrix \(P\) that multiplies \((x, y, z)\) to give \((y, z, x)\). Find the Matrix \(Q\) that multiplies \((y, z, x)\) to bring back \((x, y, z)\).

Question 3

(a) What 2 by 2 matrix \(R\) rotates every vector by \(90\,^{\circ}\)? \(R\) times \(\begin{pmatrix}x \\ y\end{pmatrix}\) is \(\begin{pmatrix}y \\ -x\end{pmatrix}\) (b) What 2 by 2 matrix \(R^{2}\) rotates every vector by \(180\,^{\circ}\)?

Answers

Answer 1

\[Ax = \begin{pmatrix}2 & 3 & 1 & 5\end{pmatrix}\begin{pmatrix}x \\ y \\ z \\ t \\ \end{pmatrix}\]

how many rows?

one row.

Answer 2

Find the matrix \(P\) that multiplies \((x, y, z)\) to give \((y, z, x)\).

\[\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\begin{pmatrix}x \\ y \\ z\end{pmatrix} = \begin{pmatrix}y \\ z \\ x\end{pmatrix}\] \[\therefore P = \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}\]

Find the Matrix \(Q\) that multiplies \((y, z, x)\) to bring back \((x, y, z)\).

\[\begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\begin{pmatrix}y \\ z \\ x\end{pmatrix}\] \[\therefore Q = \begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\]

Answer 3

(a)

\[\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}y \\ -x\end{pmatrix}\] \[\therefore R = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\]

(b)

Intuitively, if a \(90\,^{\circ}\) rotation occurs when R is applied once, wouldn’t a \(180\,^{\circ}\) rotation occur when R is applied twice?

\[\therefore \text{ Let's apply R to the answer obtained from (a) again.}\] \[\begin{pmatrix}y \\ -x\end{pmatrix}\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} = \begin{pmatrix}-x \\ -y\end{pmatrix}\]

\[\text{By the way, It is the same as:}\] \[\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}-x \\ -y\end{pmatrix}\]

\[\text{Here, }\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix} = -I\] \[\therefore R^{2} = -I\]