I was having some trouble with some practice problems. Please add a brief explanation with steps.

()Find a basis for the subspace of vectors in R 5 that satisfy the equation

x1 + 2x2 + 3x3 + 4x4 + 5x5 = 0.

()Let A be the 3 6 matrix given below:

A = 15552 6480 2808 1260 8358 275

0 1296 1080 684 8166 211

15552 10368 6048 3312 9528 908

Find invertible matrices P and Q such that P AQ is a diagonal matrix with only 1s and 0s along the diagonal.

() Find the 3 3 matrix M such that for each column vector v R 3 , Mv is vector that results by rotating v 45 degrees counter-clockwise about the line through the origin that is perpendicular to the plane 3x + 4y 12z = 0.

() Find the 3 3 matrix N such that for each column vector v R 3 , Nv is vector that results by reflecting v across the plane 3x + 4y 12z = 0.