Let A be an m × n matrix and u a vector in Rn. Suppose that Au = 0 (where 0 is the zero vector in Rm). Show that for any scalar c, A(cu) is also 0.

Let A be an m x n matrix, v be a vector in R", and c be a scalar. Show that A(cv) = c(Av).

Find orthonormal vectors 91, 92, and q3 as combinations of the columns of matrix A, where [2 2 Then write A as QR. A - 0 0 0 3 3 6

Let y = 0 and u = Into A enter UTU. Into B enter UUT. Into Center projwy. Into D enter (UUT)y. T let U be the 2x1 matrix whose only column is u, and let W span(u).

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. 0 --[142-9] A = -5 A nonzero vector in Nul A is (Type an integer or decimal for each matrix element.) A nonzero vector in Col A is. (Type an integer or decimal for each matrix element.)

Let A= {a;.a2,a3} and B = {b,.b2.b3} be bases for a vector space V, and suppose a, = 5b, - – 2b3. a. Find the change-of-coordinates matrix from A to B. b. Find (xlg for x= 5a, + 6a2 + az. a P B-A b. xlg =0 (Simplify your answer.)

Let V be the vector space of all (2 × 2) matrices. Determine whether the followingmatrices A, B, and C are linearly independent or dependent?A =(1 23 1 ), B =(3 −12 2 ), C =(1 −5−4 0 )