PX153 - K11 - matrix operation on vectors

the representation $A \vec{x} = \vec{b}$ describes a mapping of the vector space, $\vec{x}$ , to the vector space, $\vec{b}$ , by the matrix, $\vec{A}$

these spaces may or may not have the same dimensions
- if $det A = 0$ , or if $A_{m \times n}, m \neq n :$ the dimensions will be different
an important set of vectors of the mapping with matrix $A$ (in $n \times n$ case only) is the set of eigenvectors (from equation $: A \vec{x} = λ \vec{x}$ )
if there is a change in the coordinate system for the vectors, matrix $A$ will have different elements for the same transformation, but matrix $A$ will have the same eigenvector, eigenvalue, and determinant

examples of mapping

rotation in 2D

consider the effect of rotation about the $z$ -axis:

\hat{x} = [\begin{matrix} 1 \\ 0 \end{matrix}] \to [\begin{matrix} \cos θ \\ \sin θ \end{matrix}], \hat{y} = [\begin{matrix} 0 \\ 1 \end{matrix}] \to [\begin{matrix} - \sin θ \\ \cos θ \end{matrix}]

{\vec{r}}^{'} = x [\begin{matrix} \cos θ \\ \sin θ \end{matrix}] + y [\begin{matrix} - \sin θ \\ \cos θ \end{matrix}] = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]

matrix $R (θ)_{2 D}$ describes rotation by $θ$ about the $z$ -axis in the anti-clockwise direction as viewed from above

rotation in 3D about z-axis (anti-clockwise)

{\vec{r}}^{'} = R (θ_{z}) \vec{r} = [\begin{matrix} \cos θ_{z} & - \sin θ_{z} & 0 \\ \sin θ_{z} & \cos θ_{z} & 0 \\ 0 & 0 & 1 \end{matrix}] \vec{r} = [\begin{matrix} x \cos θ_{z} - y \sin θ_{z} \\ x \sin θ_{z} + y \cos θ_{z} \\ z \end{matrix}]

rotation about y-axis in 3D (anti-clockwise)

{\vec{r}}^{'} = R (θ_{y}) \vec{r} = [\begin{matrix} \cos θ_{y} & 0 & \sin θ_{y} \\ 0 & 1 & 0 \\ - \sin θ_{y} & 0 & \cos θ_{y} \end{matrix}] \vec{r}

rotation about x-axis in 3D (anti-clockwise)

{\vec{r}}^{'} = R (θ_{x}) \vec{r} = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos θ_{x} & - \sin θ_{x} \\ 0 & \sin θ_{x} & \cos θ_{y} \end{matrix}] \vec{r}

note:
- $R (θ_{x}), R (θ_{y}), R (θ_{z})$ can represent any arbitrary rotation in 3D
- $det R (θ_{x}), det R (θ_{y}), det R (θ_{x}) = 1$

stretching/shrinking of vectors

{\vec{r}}^{'} = [\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} α & 0 \\ 0 & α \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} α x \\ α y \end{matrix}]

- $det A = α^{2}$

inversion

$\vec{r} \to {\vec{r}}^{'} = - \vec{r} :$

[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} - x \\ - y \end{matrix}]

- $det A = 1$

the role of a determinant of a matrix is to give the magnification factor for the mapping of $A$ applied to a vector space
proof in 2D
- consider the effect of a matrix, $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]$ , on a square, $(0, 0), (0, 1), (1, 1), (1, 0)$
$\begin{align*} \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix} &= \begin{bmatrix}a_{11} \\ a_{21}\end{bmatrix} \\ \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix} &= \begin{bmatrix}a_{12} \\ a_{22}\end{bmatrix} \\ \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\begin{bmatrix}1 \\ 1\end{bmatrix} &= \begin{bmatrix}a_{11}+a_{12} \\ a_{21}+a_{22}\end{bmatrix}$
- area of the new shape $= (a_{11}, a_{21}, 0) \times (a_{12}, a_{22}, 0) = (a_{11} a_{22} - a_{21} a_{12}) = det A$