Question

explain why the columns of A^2 span Rn whenever the columns of an nxn matrix A are linearly independeny

Answer 1

Since the columns of a matrix being linearly independent is equivalent to the kernel (or null space, depending on text/class) being trivial, its probably better if you think about the kernel of A^2. Assume that:\[A^2x=0\]This is the same as:\[A(Ax)=0\]Since the columns of A are linearly independent, the kernel of A is trivial. Hence:\[A(Ax)=0\Longrightarrow Ax=0\]Again, since the kernel of A is trivial:\[Ax=0\Longrightarrow x=0\]So we have shown that the kernel of A^2 is trivial, which implies that the columns of A^2 are linearly independent.

Answer 2

thank you so much!