Lee R. answered • 01/20/21
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If column vectors of A are linearly independent, then determinant of matrix A is nonzero.
By properties of determinants, the determinant of A is equivalent to the determinant of A transpose.
This implies determinant of A transpose is also nonzero.
Since the columns of A transpose are the rows of A, it follows that the rows of A are also linearly independent.
If you have not yet covered determinants and their properties then let me know.
Lee R.
Yes, to say the column vectors are linearly independent is to say the determinant is nonzero; these two statements are equivalent because theory says "column vectors are linearly independent if and only if determinant is nonzero". Because you're given that the columns are already linearly independent, this automatically implies the determinant is nonzero, bringing you closer to the statement you're trying to prove. Concerning your second statement "determinant is non-zero, when row vectors are linearly independent", is equivalent to saying "IF row vectors are linearly independent THEN the determinant is non-zero" This statement is assuming what you're trying to prove
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01/20/21
Ashley P.
Can we say determinant is non-zero, only if column vectors are linearly independent? Can't we say the determinant is non-zero, when row vectors are linearly independent?01/20/21