Ples show all steps Thank you Find two vectors u u1 u2 u3 u

Ples show all steps! Thank you.

Find two vectors u = (u_1, u_2, u_3, u_4) and v = (v_1, v_2, v_3, v_4) whose span is the kernel of the matrix [7 2 3 1 5 1 36 11]. {(u_1, u_2, u_3, u_4), (v_1, v_2, v_3, v_4)} ={(-2,3,1,0),(-11,13,0,1)}

Solution

Denote the given matrix by: A

Let, x=(x1,x2,x3,x4)\'

where \' denotes transpose

Ax=0 gives

7x1+3x2+5x3+36x4=0           ------------------>P

2x1+x2+x3+11x4=0               ------------------>Q

P-3Q gives:

x1+2x3+3x4=0

x1=-2x3-3x4

Substituting this in Q gives

2(-2x3-3x4)+x2+x3+11x4=0

x2-3x3+5x4=0

x2=3x3-5x4

So,

x=(-2x3-3x4,3x3-5x4,x3,x4)=x3(-2,3,1,0)+x4(-3,-5,0,1)

So two vectors which span kernel of matrix are:

u=(-2,3,1,0)

v=(-3,-5,0,1)