Question

Chapter 4, Section 4.6, Question 24 1 0 0 The matrix P-0 5 4 is the transition matrix from what basis 8- V1 V2, V3 to the basis (1,1,1),(1,1,0),(1,0,0)) for R*? 0 3 3 Click here to enter or edit your answer V1 Click here to enter or edit your answer 12 -(?? D Click here to enter or edit your answer V3 Click if you would like to Show Work for this questinen Show Work

Answer 1

Let v₁ = (x₁,x₂,x₃)^T, v₂ =( y₁,y₂,y₃)^T and v₃ =( z₁,z₂,z₃)^T. Also let A =

1	1	1	x1	y1	z1
1	1	0	x2	y2	z2
1	0	0	x3	y3	z3

The RREF of A (on using the RREF calculator at https://www.emathhelp.net/calculators/linear-algebra/reduced-row-echelon-form-rref-caclulator/) is

1	0	0	x3	y3	z3
0	1	0	x2- x3	y2- y3	z2- z3
0	0	1	x1-x2	y1-y2	z1-z2

Hence

x3	y3	z3
x2- x3	y2- y3	z2- z3
x1-x2	y1-y2	z1-z2

=

1	0	0
0	5	4
0	3	3

Then, we have x₃ = 1, x₂-x₃ = 0, x₁-x₂=0 so that x₂ = 1,and x₁ = 1.

Also, y₃ = 0, y₂- y₃= 5 and y₁-y₂ =3 so that y₂ =5 and y₁ =8.

and z₃ = 0, z₂- z₃ = 4 and z₁-z₂ =3 so that z₂ =4 and z₁ =7.

Thus, v₁ = (1,1,1)^T, v₂ =(8,3,0)^T and v₃ =(7,4,0)^T .