GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 16

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) This is not always true...suppose rank is 5 and dim(Col(A)) = 2 then nul space dimension will be 3.

B) Not true because columns of A are in R11.

C) This is also not true because the solution of Ax = b is of the form:

 

x = (constant vector) + (combination of LI vectors) 

due to constant vector, the solution is not a subspace, it just lies in R5.

D) This is true Ax=0 has infinitely many solutions, then cols of A are LD so rank(A)<5.