Question 1

+3−0Single correct

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined as:

$T(x,y,z) = (x + y - z, ay + 2bz, x - y + z)$

Let $A$ be the matrix representation of $T$ with respect to the standard bases of $\mathbb{R}^3$ and let $B = \begin{bmatrix} 3 & 0 & 3 \\ 1 & 0 & 1 \\ 1 & 0 & 3 \end{bmatrix}$ .

Consider the following statements:

S1: If $A$ and $B$ are similar, then the range of the linear transformation $T$ is a plane in $\mathbb{R}^3$ .

S2: If $a = 2$ and $b = -1$ , then $A$ and $B$ are equivalent matrices, but not similar.

Which of the following options is true?

A.
S1 is true and S2 is false.
B.
S1 is false and S2 is true.
C.
Both S1 and S2 are true.
D.
Both S1 and S2 are false.

Question 1

+3−0Single correct

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined as:

$T(x,y,z) = (x + y - z, ay + 2bz, x - y + z)$

Let $A$ be the matrix representation of $T$ with respect to the standard bases of $\mathbb{R}^3$ and let $B = \begin{bmatrix} 3 & 0 & 3 \\ 1 & 0 & 1 \\ 1 & 0 & 3 \end{bmatrix}$ .

Consider the following statements:

S1: If $A$ and $B$ are similar, then the range of the linear transformation $T$ is a plane in $\mathbb{R}^3$ .

S2: If $a = 2$ and $b = -1$ , then $A$ and $B$ are equivalent matrices, but not similar.

Which of the following options is true?

A.
S1 is true and S2 is false.
B.
S1 is false and S2 is true.
C.
Both S1 and S2 are true.
D.
Both S1 and S2 are false.

Question 1

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Question 1

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