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?
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?
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For a matrix \( M \), the rank tells how many of its columns (or rows) are linearly independent.
Updated On: Dec 3, 2025
- If rank(\(M\)) = 1, then \(\{Me_1, Me_2\}\) is a linearly independent set
- If rank(\(M\)) = 2, then \(\{Me_1, Me_2\}\) is a linearly independent set
- If rank(\(M\)) = 2, then \(\{Me_1, Me_3\}\) is a linearly independent set
- If rank(\(M\)) = 3, then \(\{Me_1, Me_3\}\) is a linearly independent set
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The Correct Option is D
Solution and Explanation
Step 1: Understand the meaning.
Rank of \( M \) = dimension of its column space = maximum number of linearly independent columns.
Step 2: Each \( Me_i \) is the \( i^{th} \) column of \( M \).
Hence, \( \{Me_1, Me_2, Me_3\} \) are the columns of \( M \).
Step 3: If rank(\(M\)) = 2,
Then any two columns of \( M \) can be linearly independent.
Thus, \( \{Me_1, Me_2\} \) form a linearly independent set.
Final Answer: (B)
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