Question

1) solving the following system of linear equations:

x1+ 2x2 + 3x3 + 4x4 + 5x5= 6

x1+ 2x2 + 4x3 + 3x4 + 7x5= 5

x1+ 2x2 + 2x3 + 5x4 + 4x5= 9

Write the solution in parametric form.

2) Define a function T:R^(3) to R^(2) by T(x,y,z) = (x + y + z,x +2y - 3z)

a) Show that T is a linear transofrmation

b) Find all the vectors in the kernel of T

c) Show that T is onto

d) Find the matrix representation of T relative to the standardbasis of R^(3) and R^(2)

3) Show that B= {(1,1,1), (1,1,0), (0,1,1)} is a basis forR^(3). Find the coordinate vector of (1,2,3) relative to the basisof B.

4) Use the Gram-Schmidt process to find an orthonormal basis forthe subspace of R^(4) spanned by the vectors(1,0,1,1), (1,0,1,0), (0,0,1,1).

5) Consider the Matrix B = {(3,2,-3), (-3,-4,9), (-1,-2,5)}.Find an invertible matrix P and a diagonal matrix D such thatB=PDP^(-1).

ALL WORK AND STEPS MUST BE SHOWN OR YOUR RESPONSE WILL NOT GETRATED!!