Question

For a real \(4 \times 3\) matrix \(M\) and the standard basis \(\{e_1,e_2,e_3\}\) of \(\mathbb{R}^3\), which of the following statements is/are true?
(I) If \(\text{rank}(M)=1\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(II) If \(\text{rank}(M)=2\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(III) If \(\text{rank}(M)=3\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).

• A. Only (III)
• B. Both (I) and (II)
• C. Both (I) and (III)
• D. Both (II) and (III)

Hint:

Rank r means r independent columns; check whether the first two columns must be among them.

Answer 1

The Correct Option is A

Step 1: Note that \(Me_1\) and \(Me_2\) are simply the first and second columns of M, call them \(c_1\) and \(c_2\). The question is really about when the first two columns of a rank-r matrix must be linearly independent.

Step 2: Statement (I): rank(M) = 1 means the column space of M is 1-dimensional, so every column of M is a scalar multiple of a single fixed vector. In particular \(c_1\) and \(c_2\) both lie in this 1-dimensional space, so \(\{c_1,c_2\}\) is automatically linearly dependent. Statement (I) is false.

Step 3: Statement (II): rank(M) = 2 means the three columns \(c_1,c_2,c_3\) together span a 2-dimensional space, but this does not force any particular pair, such as \(\{c_1,c_2\}\), to be independent. For a counterexample, let \(c_1 = c_2 = (1,0,0,0)^T\) and \(c_3 = (0,1,0,0)^T\) be the columns of M. Then the column space of M is spanned by \((1,0,0,0)^T\) and \((0,1,0,0)^T\), so rank(M) = 2, but since \(c_1=c_2\), the set \(\{Me_1,Me_2\}\) is linearly dependent. This counterexample shows statement (II) is false in general.

Step 4: Statement (III): rank(M) = 3 for a \(4\times3\) matrix means the columns \(c_1,c_2,c_3\) are linearly independent, equivalently M is injective as a map \(\mathbb{R}^3 \to \mathbb{R}^4\). Any subset of a linearly independent set is also linearly independent, so \(\{c_1,c_2\}\) is independent. Statement (III) is always true.

Step 5: Only statement (III) is true in general. \[\boxed{\text{Only (III)}}\]