QuestionGiven $A=\left[\begin{array}{ccc}\frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right]$ , $u=\left[\begin{array}{r}9 \\ 12 \\ -15\end{array}\right]$ , find $T(u)$ and $T(v)$ for $T(\mathbf{x})=A \mathbf{x}$ .

Studdy Solution

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 $(v)$ .
$(v) = \left[\begin{array}{ccc}\frac{1}{3} &0 &0 \\0 & \frac{1}{3} &0 \\0 &0 & \frac{1}{3}\end{array}\right] \cdot \left[\begin{array}{l}h \\ c \\ d\end{array}\right]$

STEP 7

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 $(v) = \left[\begin{array}{r}\frac{h}{3} \\ \frac{c}{3} \\ \frac{d}{3}\end{array}\right]$

Was this helpful?

QuestionGiven $A=\left[\begin{array}{ccc}\frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right]$ , $u=\left[\begin{array}{r}9 \\ 12 \\ -15\end{array}\right]$ , find $T(u)$ and $T(v)$ for $T(\mathbf{x})=A \mathbf{x}$ .

Studdy Solution

STEP 1

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 $(v)$ .
$(v) = \left[\begin{array}{ccc}\frac{1}{3} &0 &0 \\0 & \frac{1}{3} &0 \\0 &0 & \frac{1}{3}\end{array}\right] \cdot \left[\begin{array}{l}h \\ c \\ d\end{array}\right]$

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

To find (u)(u)(u), we need to multiply the matrix AAA by the vector uuu. 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⋅u(u) = A \cdot u(u)=A⋅u

STEP 3

STEP 4

STEP 5

To find (v)(v)(v), we need to multiply the matrix AAA by the vector vvv. 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⋅v(v) = A \cdot v(v)=A⋅v

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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.
$(u) = A \cdot u$

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.
$(v) = A \cdot v$