GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 14

Question

Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are true?

• A. $\operatorname{dim}(\operatorname{Col} A) \geq \operatorname{dim}(\operatorname{Nul} A)$.
• B. If the columns of $A$ are linearly independent, then the range of $T$ is $\mathbf{R}^{5}$.
• C. Suppose $b$ is a vector so that the matrix equation $A x=b$ is consistent. Then the set of solutions to $A x=b$ must be a subspace of $\mathbf{R}^{5}$.
• D. If the matrix equation $A x=0$ has infinitely many solutions, then $\operatorname{rank}(A) \leq 4$.

Answer 1

A. False, 

More than 2 Linearly Dependent column can also exist in A.

 

B. False,

domain of Ax =  Rⁿ = 5 

co-domain of Ax = R^m = 11

If columns are Linearly Independent, then range of T will be Col(A) in subset of Co-domain i.e, R^m = R¹¹

 

C. False

Null(A) will be in Rⁿ = R⁵, and b = linear combination of Columns of A.

for Ax = b will have solution like, 

b = x₁[] + x₂[] + x₃[] + x₄[] + x₅[] , not all x_i = 0, 

a constant term b is added, so not a subspace.

 

D. True

m = 11, n = 5

m > n,

rank(A) <= n, for existence of infinitely many solutions rank < n, such that atleast one free variables could exist.