Question

Let B_1={u_1,u_2,u_3} and B_2 = {v_1,v_2,v_3} be the bases for R^3 in which u_1 =(-3,0,-3) u_2=(-3, 2, -1) u_3=(1,6,-1) v_1=(-6, -6,0) v_2=(-2,-6,4) and v_3=(-2,-3,7). find the coordinate matrix with respect to B_1 of w=(-5,8,-5) and then use the transition matrix to find [w]_B_2

Answer 1

ah scary

Answer 2

Such a long question but yeah I already found the transition matrix

Answer 3

[.75 .75 1/12; -.75 -17/12 -17/12; 0 2/3 2/3]

Answer 4

that looks familiar...was that the one we worked out a day or two ago?

Answer 5

no, did someone ask this?

Answer 6

don't know how to find the coordinate matrix with respect to be B_1, so is the w given for B_1?

Answer 7

i remember getting a matrix just like that, i have it written on my board lol.

Answer 8

Haha well it wasn't me probably someone from my class since we have a final tmr!

Answer 9

could you explain how to do it?

Answer 10

How to get that matrix?

Answer 11

yeah the coordinate matrix, I actually don't really understand what it's asking

Answer 12

im thinking that vector w is in the standard basis right now. and you want to find its coordinates with respect to B_1.

Answer 13

So you would set up an augmented matrix to find out what combination of u1, u2 and u3 makes w.

Answer 14

in the answer key it says [w]_B_1 =....blahblah

Answer 15

-3 -3 1 -5
0 2 6 8
-3 -1 -1 5

Answer 16

so are we finding [w]_B_1?

Answer 17

right. you are trying to solve:
\[c_1u_1+c_2u_2+c_3u_3 = w\]
You want those c's, those are the coordinates with respect to the basis B_1.

Answer 18

and you're given w ?

Answer 19

so you use the w=(-5, 8,-5)

Answer 20

right. the w given is that vector (-5, 8, -5)

Answer 21

how would you do this? w dot product with each v?

Answer 22

oh I meant u*

Answer 23

you would just reduce that matrix i posted above, that augmented matrix. Doing dot products would only work if we had an orthogonal basis, but this isnt orthogonal.

Answer 24

okay so I got 1 0 0 -3.166; 0 1 0 4.75; 0 0 1 -0.25

Answer 25

but answer is [1; 1 ;1]?

Answer 26

Oh im sry, that very last entry is supposed to be a -5 from w.
-3 -3 1 -5
0 2 6 8
-3 -1 -1 -5

should reduce to the correct answer.

Answer 27

oh okay so is that you do if it's not orthogonal? so if I want to find [w]_B_2

Answer 28

I would put the v's in a matrix but I can't use the -5,8,-5 as the last column right

Answer 29

you are right, one way to do it is to but the v's in a matrix as columns, and w as (-5, 8, 5) in an augmented matrix.

Another way to do it is to use that change of basis matrix, the one you had posted at the very tor (the one i said i had seen before).

That matrix is a change of basis matrix from B_1 to B_2. So w in B_1 is (1,1,1). You take that matrix and multiply it with (1, 1, 1) and that should be you answer as well.

Answer 30

Ohhh it works, why can you use -5 8 -5?

Answer 31

for both augmented matrix?

Answer 32

because thats in the standard basis.

Answer 33

so the standard basis never changes

Answer 34

if you have something in the standard basis, you can convert to any other basis using an augmented matrix. if you are in some other basis (B_1) and want to convert to another basis (B_2) you need a change of basis matrix.

Answer 35

okay what if you're not given that can you find it?

Answer 36

im not sure i understand. what if your not given what?

Answer 37

the w standard basis

Answer 38

sorry for all these questions just trying to understand what I solved etc.

Answer 39

oh, if its not in standard basis, then they need to give you the basis its in. Say we were given w is (1, 1, 1) with respect to B_1 and asked what is w with respect to B_2 (we were never told what w is in standard basis).
We would have to calculate the change of basis matrix and multiply w by that matrix.

Answer 40

ohh okay I get it so you'd need to know the [w]_B_2

Answer 41

oh wait nvm sorry I don't get this statement 'We would have to calculate the change of basis matrix and multiply w by that matrix.'

Answer 42

since you don't know w and you don't know the answer to w * change of baiss

Answer 43

basis*

Answer 44

you would need to know w. the problem doesnt make too much sense if you dont know w. If you didnt know w, then all you could do is create the change of basis matrix, call it A and say [w]_B2 = A[w]_B1

Answer 45

Haha okay sorry for the confusion. I just really don't get coordinate basis. But thanks for clearing it up.