Change of Basis Matrices for B1 and B2 in Vector Space V - Homework Solution

[TiberiusK]

Homework Statement

Let B1 = {v1; v2; v3} be a basis of a vector space V and let B2 = {w1;w2;w3} where
w1 = v2 + v3 ; w2 = v1 + v3 ; w3 = v1 + v2
Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2
and from B2 to B1. *Use the appropriate change of basis matrix to express the vector
av1 + bv2 + cv3 as a linear combination of w1 , w2 and w3 .

Homework Equations

The Attempt at a Solution

I need some help on getting started with the last part*

 

[AkilMAI]

do you mean this part?...Use the appropriate change of basis matrix to express the vector
av1 + bv2 + cv3 as a linear combination of w1 , w2 and w3?

 

[vela]

You presumably found the change-of-basis matrices. Ask yourself, what exactly do these matrices do? If you can understand that, the last part is trivial.

 

[TiberiusK]

You presumably found the change-of-basis matrices. Ask yourself, what exactly do these matrices do? If you can understand that, the last part is trivial.

there are used to change different objects of the space to the new basis,a quick and easy way to convert between the coordinate vectors from one basis to a different basis...so I use the change of basis matrix to express form B2 toB1 to express that vector?
av1 + bv2 + cv3=c1w1+c2w2+c3w3?

 

[vela]

there are used to change different objects of the space to the new basis, a quick and easy way to convert between the coordinate vectors from one basis to a different basis

That's the essential point. A single vector has different representations depending on the bases you're using. The matrices allow you to convert between these representations easily and quickly.

...so I use the change of basis matrix to express form B2 toB1 to express that vector?
av1 + bv2 + cv3=c1w1+c2w2+c3w3?

Sorry, I have no idea what your question means.

What's the coordinate vector for x=av₁+bv₂+cv₃ relative to the {v_i} basis? Multiply it by the appropriate matrix, and you will get the coordinate vector for x relative to the {w_i} basis.

 

[TiberiusK]

Sorry,but there is a reason my question is incomprehensible...I don't understand the entire sentence from the last part of the exercise...I found the change of basis matrices...but this "av1 + bv2 + cv3 as a linear combination of w1 , w2 and w3 ".is the confusing part...ok last questions(if I'm incoherent is because I'm tired)
x=av1+bv2+cv3...x is {w1,w2,w3}?...and "Multiply it by the appropriate matrix"?you mean the change of basis matrix?
Thanks

 

[vela]

OK, start with some vector x in vector space V and a basis {v_i} of V. You can express x as a linear combination of the basis vectors:

x = av₁+bv₂+cv₃.

The 3-tuplet (a,b,c) is the coordinate vector of x relative to the basis {v_i}. But now you have a second basis {w_i} of V, and you can similarly express x as a linear combination of this second set of basis vectors:

x = dw₁+ew₂+fw₃.

The 3-tuplet (d,e,f) is the coordinate vector of x relative to the basis {w_i}.

Though (a,b,c) and (d,e,f) will generally be two different triplets of numbers, they both represent the vector x. You have two different representations for the same vector because you have two different bases. A change-of-basis matrix allows you to convert from one representation, e.g. (a,b,c), to another, e.g. (d,e,f).

 

[TiberiusK]

Thank you vela..I understand completely:)