In a 3x5 matrix A, the columns cannot be linearly independent due to the rank limitation. The maximum rank of A is 3, which corresponds to the maximum number of linearly independent columns. Since A has 5 columns, this guarantees that at least some columns must be linearly dependent. Additionally, the rank also reflects the number of linearly independent rows, reinforcing the conclusion that linear independence is not possible with more columns than the rank.
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If A is a 3x5 matrix, explain why the columns of A must
be linearly independent.
Correct! (Yes) (You might perhaps have added that the rank is also the number of linearly independent rows in A, which is why the rank is at most 3.)Yuuki said:i thought if A is 3x5, the columns of A must be linearly dependent, since
the rank is at most 3, and the rank is the number of linearly independent columns in A.
but there are 5 columns in A, so the columns of A must be linearly dependent :/