- #1
Eleni
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Homework Statement
Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and
B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)}
a) Show that B1 is a basis for [R][/3]
b) Find the coordinates of w=(2,3,1) relative to B1
c)Given that B2 is a basis for [R[/3], find the transition matrix [P[/B1→B2]
d) Use the transition matrix [P[/B1→B2] to find the coordinates of w relative to B2
e) What is the relationship between [P[/B1→B2] and [P[/B2→B1]? Find [P[/B2→B1]
f) Suppose [[x]][/B2]= (0, 3, -1). Use the appropriate transition matrix to find [[x]][/B1]
The Attempt at a Solution
a) if B1 is a basis for [R][/3] then there should be a unique solution for all the coefficients where;
[x][/1] [u][/1] + [x][/2] [u][/2] + [x][/3][u][/3]=(0,0,0)
the system of equation gives the matrix of the form;
1 0 1 1 0 0
1 2 0 0 1 0
1 -1 2 →RREF = 0 0 1
solving for Ax=0 gives [x][/1] [u][/1] + [x][/2] [u][/2] + [x][/3][u][/3]=(0,0,0)
Thus a unique solution exists for all R(X)∈[R][/3] and B1 is a basis for [R][/3]
b) To find w relative to B1 we use the equation from part a) and sub in w values.
[[w]][/B1] = [x][/1] (1,1,1) + [x][/2] (0,2,-1) + [x][/3](1,0,2)=(2,3,1)
This is matrix form yeilds;
1 0 1 2 1 0 0 3
1 2 0 3 0 1 0 0
1 -1 2 1→RREF = 0 0 1 -1
∴ [x][/1] = 3
[x][/2] = 0
[x][/3] = -1
So the coordinate vector [[w]][/B1] = (3, 0, -1)
c) The transition matrix [P[/B1→B2] =
1 1 0 1 0 1 1 0 0 2/3 2/3 1/6
1 -1 2 1 2 0 0 1 0 1/3 -2/3 5/6
0 2 1 1 -1 2 →RREF = 0 0 1 1/3 1/3 1/3
∴ The transition matrix [P[/B1→B2] = 2/3 2/3 1/6
1/3 -2/3 5/6
1/3 1/3 1/3
d) To find [[w]][/B2] we will callculate [[P[/B1→B2]][w]
2/3 2/3 1/6 2 1 0 0 5
1/3 -2/3 5/6 3 0 1 0 -2
1/3 1/3 1/3 1 →RREF = 0 0 1 0
∴ The coordinate vector of [[w]][/B2] = (5,-2,0)
e)What is the relationship between [P[/B1→B2] and [P[/B2→B1]? Find [P[/B2→B1]
I am unsure about this one. I am sure there must be a theorem or rule regarding this but I can't find it in my textbook.
f) I am guessing that once I have found [P[/B2→B1] then I can find [[x][/B1] the same as I found [[w]][/B2] but with the transition matrix from [P[/B2→B1] in part e)I am not 100% confident in my answers part parts a) through to d) so if you can see errors please correct me. However I am predominantly concerned with parts e) and f)
I thank you in advance for any advice and help.