Questions: Let A = [ [1/4, 0, 0], [0, 1/4, 0], [0, 0, 1/4] ], u = [4, 12, -8], and v = [k, f, h]. Define T: R^3 -> R^3 by T(x) = Ax. Find T(u) and T(v). T(u) = □ (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Let A = [ [1/4, 0, 0], [0, 1/4, 0], [0, 0, 1/4] ], u = [4, 12, -8], and v = [k, f, h]. Define T: R^3 -> R^3 by T(x) = Ax. Find T(u) and T(v). T(u) = □ (Simplify your answer. Use integers or fractions for any numbers in the expression.)
Transcript text: Let $A=\left[\begin{array}{lll}\frac{1}{4} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4}\end{array}\right], u=\left[\begin{array}{r}4 \\ 12 \\ -8\end{array}\right]$, and $v=\left[\begin{array}{l}k \\ f \\ h\end{array}\right]$. Define $T: R^{3} \rightarrow R^{3}$ by $T(x)=A x$. Find $T(u)$ and $T(v)$. \[ \mathrm{T}(\mathbf{u})=\square \] (Simplify your answer. Use integers or fractions for any numbers in the expression.)
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Solution

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Solution Steps

Step 1: Compute \( T(u) \) by multiplying matrix \( A \) with vector \( u \)

Given: \[ A = \begin{bmatrix} \frac{1}{4} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} \end{bmatrix}, \quad u = \begin{bmatrix} 4 \\ 12 \\ -8 \end{bmatrix} \] The transformation \( T(u) \) is calculated as: \[ T(u) = A \cdot u = \begin{bmatrix} \frac{1}{4} \cdot 4 + 0 \cdot 12 + 0 \cdot (-8) \\ 0 \cdot 4 + \frac{1}{4} \cdot 12 + 0 \cdot (-8) \\ 0 \cdot 4 + 0 \cdot 12 + \frac{1}{4} \cdot (-8) \end{bmatrix} \]

Step 2: Simplify the multiplication for \( T(u) \)

\[ T(u) = \begin{bmatrix} 1 + 0 + 0 \\ 0 + 3 + 0 \\ 0 + 0 - 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]

Step 3: Compute \( T(v) \) by multiplying matrix \( A \) with vector \( v \)

Given: \[ v = \begin{bmatrix} k \\ f \\ h \end{bmatrix} \] The transformation \( T(v) \) is calculated as: \[ T(v) = A \cdot v = \begin{bmatrix} \frac{1}{4} \cdot k + 0 \cdot f + 0 \cdot h \\ 0 \cdot k + \frac{1}{4} \cdot f + 0 \cdot h \\ 0 \cdot k + 0 \cdot f + \frac{1}{4} \cdot h \end{bmatrix} \]

Step 4: Simplify the multiplication for \( T(v) \)

\[ T(v) = \begin{bmatrix} \frac{k}{4} \\ \frac{f}{4} \\ \frac{h}{4} \end{bmatrix} \]

The remaining questions are left unanswered as per the guidelines.

Final Answer

\(\mathrm{T}(\mathbf{u}) = \boxed{\begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix}}\)
\(\mathrm{T}(\mathbf{v}) = \boxed{\begin{bmatrix} \frac{k}{4} \\ \frac{f}{4} \\ \frac{h}{4} \end{bmatrix}}\)

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