Is the below statement True False or Sometimes True Suppose

Is the below statement True, False, or Sometimes True: Suppose A is a matrix in R^m times n. Suppose that A has orthonormal columns. Then the columns of A are linearly independent. True False Sometimes True: The columns of .4 are linearly independent only when m = n, i.e. when .4 is a square matrix.

Solution

sometimes true: if A is a suqare matrix.

we know that a square matrix with orthonormal colums satisfies AA^T = I. Let v₁,v₂,v₃.....v_n be thr rows of A or the columns of A^T. then AA^T = I says < v _i, v_j > = 0, if i not equals to j and < v _i, v_j > = 1, if i = j.