GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 11

Question

Which of the following statements are true?

• A. There exists a $3 \times 3$ matrix $A$ and vectors $b, c \in \mathbb{R}^{3}$ such that the linear system $A x=b$ has a unique solution but $A x=c$ has infinitely many solutions.
• B. If $\left\{v_{1}, v_{2}, v_{3}\right\}$ is a basis for $\mathbb{R}^{3}$ then so is $\left\{v_{1}, v_{2}, v_{1}+v_{2}+v_{3}\right\}$.
• C. If $W$ is a subset of $\mathbb{R}^{3}$ that contains $\underline{0}$ and satisfies $v+w \in W$ for every $v, w \in W$, then $W$ is a subspace of $\mathbb{R}^{3}$.
• D. If $A$ and $B$ are $n \times n$ matrices, both of rank $n$, then the column space of $A$ is equal to the column space of $B$.

Answer 1

A. False
A linear system can't have both unique and infinitely many solutions at the same time. In Ax=b and Ax=c, if c and b are linear combination of columns of A, then the systems have a solution. Unique or Infinitely many solutions is determined by the linear dependency among the columns of A. If cols of A are LD, then Infinite sols, If LI, then Unique.

B. True.
Take v1 = [1, 0, 0], v2 = [0,1,0], v3 = [0,0,1]. Then {v1, v2, v1+v2+v3} = {[1,0,0], [0,1,0], [1,1,1]}. Still it is linearly Independent which can Span R^3

C. False
W should be closure under multiplication as well.

D. True.
Let's say A and B are 3*3 Matrix. Since both matrix have full rank, i.e, 3, both Matrix span the entire R^3. So, the column spaces of A and B have same span