Question

4. Show that B = {(1, 1, 1),(1, 1, 0),(0, 1, 1)} is a basis for R3 . Find the coordinate vector of (1, 2, 3) relative to the basis B.

Answer 1

Explanation:

consider B in matrix form

We have

`\left[\begin{array}{ccc}1&1&1\\1&1&0\\0&1&1\end{array}\right]`

Reduce this to row echelon form

by R1= R1-R3

we get

`\left[\begin{array}{ccc}1&0&0\\1&1&0\\0&1&1\end{array}\right]`

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)