MHB Understanding the Linear Independence of Columns in a 3x5 Matrix

[annie122]

If A is a 3x5 matrix, explain why the columns of A must
be linearly independent.

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 :/

 

[Opalg]

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 :/

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.)