GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 13

Question

Suppose $A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, and let $T$ be the matrix transformation $T(x)=A B x$. Which of the following must be true?

• A. The column space of $A B$ is a subspace of $\mathbf{R}^{2}$.
• B. Every vector in the null space of $A B$ is also in the null space of $A$.
• C. $T$ has domain $\mathbf{R}^{2}$ and codomain $\mathbf{R}^{4}$.
• D. $T$ cannot be onto.

Answer 1

A is 4x3, B is 3x2
AB will be 4x2, that is, $T: R^2 \rightarrow R^4$

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Option A: False

AB matrix is 4x2, that is the basis of column space will be a plane in R4

So, it is a subspace of R4 not R2

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Option B: False

Null space of A is in R3
Null space of AB is in R2

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Option C: True

Since T is R4 to R2
Domain is R2
CoDomain is R4

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Option D: True

Onto Function means that no solution is not possible. Every element in CoDomain has atleast one solution.
AB is 4x2
maximum rank possible is 2, but there are 4 rows

So, the condition for no solution, that is, [ 0 0 0 .. | non-zero ] can exist in the matrix [AB | R4}
It means Onto is not possible.

Answer 2

The associated matrix with this Linear transformation is AB its dimension will be 4x2.
Option C - AB can take input as ABX , only if X has dimension of 2xAnything , hence domain is nothing but input so , domain is in R2 and ABX will give a 4xanything output , so co-domain is nothing but output , so co-domain will be in R4.

Option D- For Onto , no solution is not acceptable hence we need to have 4 LI col in AB matrix which is in R4, but it has only 2 col , or we can say we should have pivot in every row of AB , which is not there hence not Onto.