Linear Algebra: Change of basis

In summary, the conversation discusses finding the vector AUE in a given basis and finding the transformation Uw = BUw corresponding to VE = AUE. This involves a change of basis, where A represents a linear transformation in the standard basis and B represents the same transformation in another basis. The question is asking for a similarity transformation to find the matrix that transforms vector u in the given basis to the same point as A mapped vector U in the standard basis.
  • #1
mccoy1
117
0
(a) Let A (matrix) =c1= [1,2,1], c2 = [0,1,2], c3 = [3,-2,-1] be a matrix (c1,c2,c3 refer to the columns of the matrix A, which is a 3x3 matrix) expressed in the standard basis and let w1 = (0,0,1)T, w2 = (0,1,2)T , w3 =(3,0,2)T , find the vector AUE
 in w basis.
(b). Referring to problem (a), find the transformation Uw = BUw, corresponding to VE = AUE.

The Attempt at a Solution



(a) Ans = W^-1A UE...no problem.
(b), I think I'm either stupid or the question is (lol well you know what i mean)...what do they mean by 'Uw = BUw, corresponding to VE = AUE'?
Thanks fellows.
 
Last edited:
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  • #2
What do you think guys?
 
  • #3
A is the matrix representing a linear transformation in one basis, the "standard" basis, {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B is the matrix representing the same linear transformation in another basis, {(0, 0, 1), (0, 1, 2), (3, 0, 2)}.

That's why this was titled "change of basis".
 
  • #4
HallsofIvy said:
A is the matrix representing a linear transformation in one basis, the "standard" basis, {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B is the matrix representing the same linear transformation in another basis, {(0, 0, 1), (0, 1, 2), (3, 0, 2)}.

That's why this was titled "change of basis".

Thanks for your help. I have no problem with change of basis, so to speak and that's why i did the first question (part a)...part a involved a change of basis. What I do have problem with though is what they meant by transformation Uw = BUw corresponding to VE = AUE. Your explanation makes sense, I'm starting to wonder why they just didn't simply us to find the matrix that transform vector u (in w basis) to the same point/that A mapped vector U(basis E, i.e the std basis). From your explanation, I think they are asking for a similarity transformation. If that's true, then I hate the use of the term correspond because it doesn't mean anything (in my opinion)..
Thanks.
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that can be used to represent any vector in a vector space. It is similar to the coordinate system in geometry, where a point can be represented by its coordinates on a set of axes.

2. What is a change of basis in linear algebra?

A change of basis in linear algebra is a transformation of a vector from one basis to another. It allows us to express the same vector in different bases, providing a different perspective on the vector and its properties.

3. Why is change of basis important in linear algebra?

Change of basis is important in linear algebra because it allows us to simplify complex calculations and proofs by choosing a more convenient basis. It also provides a powerful tool for understanding and analyzing vector spaces, transformations, and systems of linear equations.

4. How do you perform a change of basis in linear algebra?

To perform a change of basis in linear algebra, we use a matrix called the transition matrix. This matrix is formed by arranging the basis vectors of the new basis as columns and expressing them in terms of the old basis. To transform a vector from the old basis to the new basis, we multiply the transition matrix by the vector.

5. Can we change the basis of any vector space in linear algebra?

Yes, we can change the basis of any vector space in linear algebra as long as the new basis is also a set of linearly independent vectors. This is because a change of basis is essentially a change of perspective, and as long as the fundamental properties of the vector space are preserved, any basis can be chosen.

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