Math Snap

STEP 1

Assumptions1. Matrix $A=\left[\begin{array}{ccc}\frac{1}{3} &0 &0 \\0 & \frac{1}{3} &0 \\0 &0 & \frac{1}{3}\end{array}\right]$
. Vector $u=\left[\begin{array}{r}9 \\12 \\ -15\end{array}\right]$
3. Vector $\mathbf{v}=\left[\begin{array}{l}h \\ c \\ d\end{array}\right]$
4. The transformation $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ is defined by $(\mathbf{x})=A x$

STEP 2

To find $(u)$, we need to multiply the matrix $A$ by the vector $u$. This is done by multiplying each element of a row of the matrix by the corresponding element of the vector, and then summing these products.
$$(u) = A \cdot u$$

STEP 3

Plug in the values for the matrix $A$ and the vector $u$ to calculate $(u)$.
$$(u) = \left[\begin{array}{ccc}\frac{1}{3} &0 &0 \\0 & \frac{1}{3} &0 \\0 &0 & \frac{1}{3}\end{array}\right] \cdot \left[\begin{array}{r}9 \\12 \\ -15\end{array}\right]$$

STEP 4

Perform the matrix-vector multiplication to calculate $(u)$.
$$(u) = \left[\begin{array}{r}\frac{1}{3} \cdot9 \\ \frac{1}{3} \cdot12 \\ \frac{1}{3} \cdot (-15)\end{array}\right] = \left[\begin{array}{r}3 \\4 \\ -\end{array}\right]$$

STEP 5

To find $(v)$, we need to multiply the matrix $A$ by the vector $v$. This is done by multiplying each element of a row of the matrix by the corresponding element of the vector, and then summing these products.
$$(v) = A \cdot v$$

STEP 6

Plug in the values for the matrix $A$ and the vector $v$ to calculate $(v)$.
$$(v) = \left[\begin{array}{ccc}\frac{1}{3} &0 &0 \\0 & \frac{1}{3} &0 \\0 &0 & \frac{1}{3}\end{array}\right] \cdot \left[\begin{array}{l}h \\ c \\ d\end{array}\right]$$

Perform the matrix-vector multiplication to calculate $(v)$.
$$(v) = \left[\begin{array}{r}\frac{1}{3} \cdot h \\ \frac{1}{3} \cdot c \\ \frac{1}{3} \cdot d\end{array}\right] = \left[\begin{array}{r}\frac{h}{3} \\ \frac{c}{3} \\ \frac{d}{3}\end{array}\right]$$So, $(u) = \left[\begin{array}{r}3 \\4 \\ -5\end{array}\right]$ and $(v) = \left[\begin{array}{r}\frac{h}{3} \\ \frac{c}{3} \\ \frac{d}{3}\end{array}\right]$