Let AequalsStart 2 By 2 Table 1st Row 1st Column 3 2nd Column 0 2nd Row 1st Column 0 2nd Column 3 EndTable 
 ​, and define Upper T : set of real numbers R squared right arrow set of real numbers R squared by Upper T left parenthesis Bold x right parenthesis equals Upper A Bold x. Find the images under T of Bold u equals Start 2 By 1 Table 1st Row 1st Column 1 2nd Row 1st Column negative 5 EndTable 
  and Bold v equals Start 2 By 1 Table 1st Row 1st Column a 2nd Row 1st Column b EndTable 
 .

Let's solve this problem step by step.

We are given the matrix \( A \) and the transformation \( T \). The transformation \( T \) applies the matrix \( A \) to the vectors \( \mathbf{u} \) and \( \mathbf{v} \). First, let's define the matrix \( A \), vector \( \mathbf{u} \), and vector \( \mathbf{v} \):

\[
A = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}
\]

\[
\mathbf{u} = \begin{pmatrix} 1 \\ -5 \end{pmatrix}
\]

\[
\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}
\]

The transformation \( T(\mathbf{x}) = A\mathbf{x} \) means multiplying the matrix \( A \) by the vector \( \mathbf{x} \).

### Image of \( \mathbf{u} \) under \( T \):
We compute \( T(\mathbf{u}) \) as follows:

\[
T(\mathbf{u}) = A \mathbf{u} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ -5 \end{pmatrix}
\]

Carrying out the matrix multiplication:

\[
T(\mathbf{u}) = \begin{pmatrix} 3(1) + 0(-5) \\ 0(1) + 3(-5) \end{pmatrix} = \begin{pmatrix} 3 \\ -15 \end{pmatrix}
\]

So, the image of \( \mathbf{u} \) under \( T \) is:

\[
T(\mathbf{u}) = \begin{pmatrix} 3 \\ -15 \end{pmatrix}
\]

### Image of \( \mathbf{v} \) under \( T \):
We compute \( T(\mathbf{v}) \) as follows:

\[
T(\mathbf{v}) = A \mathbf{v} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix}
\]

Carrying out the matrix multiplication:

\[
T(\mathbf{v}) = \begin{pmatrix} 3a + 0b \\ 0a + 3b \end{pmatrix} = \begin{pmatrix} 3a \\ 3b \end{pmatrix}
\]

So, the image of \( \mathbf{v} \) under \( T \) is:

\[
T(\mathbf{v}) = \begin{pmatrix} 3a \\ 3b \end{pmatrix}
\]

### Final Answer:
- The image of \( \mathbf{u} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \) under \( T \) is \( T(\mathbf{u}) = \begin{pmatrix} 3 \\ -15 \end{pmatrix} \).
- The image of \( \mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix} \) under \( T \) is \( T(\mathbf{v}) = \begin{pmatrix} 3a \\ 3b \end{pmatrix} \).

Would you like more details or have any questions about this?

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Here are 5 related questions for further exploration:

1. What is the inverse of the matrix \( A \), and how would you find it?
2. If \( A \) were not diagonal, how would you approach the matrix multiplication?
3. What geometric transformation does \( T \) represent in \( \mathbb{R}^2 \)?
4. How does the determinant of \( A \) affect the transformation?
5. Can you find the eigenvalues and eigenvectors of the matrix \( A \)?

**Tip:** Diagonal matrices like \( A \) are particularly easy to work with because multiplying them with vectors only scales the components by the diagonal elements.

Let AequalsStart 2 By 2 Table 1st Row 1st Column 3 2nd Column 0 2nd Row 1st Column 0 2nd Column 3 EndTable , and define Upper T : set of real numbers R squared right arrow set of real numbers R squared by Upper T left parenthesis Bold x right parenthesis equals Upper A Bold x. Find the images under T of Bold u equals Start 2 By 1 Table 1st Row 1st Column 1 2nd Row 1st Column negative 5 EndTable and Bold v equals Start 2 By 1 Table 1st Row 1st Column a 2nd Row 1st Column b EndTable .

Learn how to find the images of vectors under a linear transformation using matrix multiplication. This problem involves a diagonal matrix and vectors in R^2, demonstrating concepts in linear algebra. Suitable for college-level students.

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