Let A be a 2 × 2 matrix, and let LA be the linear operator defined by LA(x) = Ax Show that (a) LA maps R2 onto the column space of A. (b) if A is nonsingular, then LA maps R2 onto R2.

Let CE Rxn be a matrix such that xCx>0 for all nonzero vectors x ER". Prove that C is nonsingular.

Find the matrix representation of the linear operator T relative to B1. Then use Theorem 15 to find [T]B2

Let T be a linear operator defined on P3 by T(a0+a1x+a2x^2)=a0+a1(x-1)+a2(x-1)^2. The third row of the matrix associated with T with respect to base 1, x, x^2 is: a) (0,0,1) b) other response c) (1,-2,1) d) (1,0,0) e) (-1,1,0)