Question

Let \( M \) be a \( 4 \times 3 \) real matrix and let \( \{e_1, e_2, e_3\} \) be the standard basis of \( \mathbb{R}^3. \) Which of the following is true?

• A. If rank(\(M\)) = 1, then \(\{Me_1, Me_2\}\) is a linearly independent set
• B. If rank(\(M\)) = 2, then \(\{Me_1, Me_2\}\) is a linearly independent set
• C. If rank(\(M\)) = 2, then \(\{Me_1, Me_3\}\) is a linearly independent set
• D. If rank(\(M\)) = 3, then \(\{Me_1, Me_3\}\) is a linearly independent set

Hint:

For a matrix \( M \), the rank tells how many of its columns (or rows) are linearly independent.

Answer 1

The Correct Option is D

Step 1: Key observation:

When $M$ multiplies the standard basis vectors, we get:

• $Me_1$ = first column of $M$
• $Me_2$ = second column of $M$
• $Me_3$ = third column of $M$

Therefore, ${Me_1, Me_2, Me_3}$ is the set of columns of $M$.

Step 2: Analysis:

The rank of $M$ equals the maximum number of linearly independent columns of $M$.

• If $\text{rank}(M) = r$, then there are exactly $r$ linearly independent columns among the three columns of $M$.

Step 3: Evaluating each option:

(A) If $\text{rank}(M) = 1$, then ${Me_1, Me_2}$ is linearly independent

If rank is 1, there is only 1 linearly independent column. Any set of 2 columns must be linearly dependent. FALSE

(B) If $\text{rank}(M) = 2$, then ${Me_1, Me_2}$ is linearly independent

If rank is 2, there are exactly 2 linearly independent columns. However, this doesn't guarantee that the first two columns are the independent ones. The independent columns could be columns 1 and 3, or columns 2 and 3. NOT NECESSARILY TRUE

(C) If $\text{rank}(M) = 2$, then ${Me_1, Me_3}$ is linearly independent

By the same reasoning as (B), this is not guaranteed. NOT NECESSARILY TRUE

(D) If $\text{rank}(M) = 3$, then ${Me_1, Me_3}$ is linearly independent

If rank is 3, all three columns are linearly independent. This means any subset of the three columns is also linearly independent. In particular, ${Me_1, Me_3}$ (columns 1 and 3) must be linearly independent. TRUE

Answer: (D) If $\text{rank}(M) = 3$, then ${Me_1, Me_3}$ is a linearly independent set