4. show that b = {(1, 1, , 1, , 1, 1)} is a basis for r3 . find the coordinate vector of (1, 2, 3) relative to the basis b.

Step-by-step explanation:

consider B in matrix form

We have

Reduce this to row echelon form

by R1= R1-R3

we get

Now R2-R1 gives Identity matrix in row echelon form.  So rank =3 hence this is a basis for R cube

to find (1,2,3) as linear combination of B

Let a, b, and c be the scalars such that

a(1,1,1)+b(1,1,0)+c(0,1,1) = (1,2,3)

Equate corresponding terms as

a+b= 1:   a+b+c =2:  a+c =3

Solving b = -1, c = 1 and a = 2

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

-7/8

step-by-step explanation:

instead of subtracting you add the -6/8 and -1/8