Matrix of Transformation (non standard basis)

In summary, the conversation discusses the definition of a linear transformation T from R3x1 to R3x1, and the subsequent steps to show that T is a linear transformation. This includes finding the matrix of T relative to the standard basis, and the matrix [T]' relative to a given basis B'. The conversation also discusses finding the transition matrix Q from B to B', and verifying that [T]' = Q[T]Q-1. The conversation ends with a clarification of some errors in the calculations for [T] and the inverse of Q.
  • #1
margaret37
12
0

Homework Statement



Define T : R3x1 to R3x1 by T = (x1, x2,x3)T = (x1, x1+x2, x1+x2+x3)T

1 Show that T is a linear transformation
2 Find [T] the matrix of T relative to the standard basis.
3 Find the matrix [T]' relative to the basis
B' = {(1,0,0)t, (1,1,0)t, (1,1,1)t
4 Find the transition matrix Q from B to B'
5 Verify that [T]' = Q[T]Q-1



Homework Equations





The Attempt at a Solution


1) I think this is done I showed that T(c(alpha + beta)) = c* T(alpha) + T(Beta) by substituting in vectors (a1, a2, a3) and (b1,b2,b3) and working throught the algebra.

2) This is what I did, I think I understood this properly:

I built a transformation Matrix [T] = T(e1) | T(e2) | T(e3)

I got
1 0 0
1 1 0
1 1 1

3) Then I think is wrong but I'm not sure

I built another matrix [T]' = T(v1) | T(v2) | t(v3)
where v1, v2, and v3 are the the vectors given above

I got
1 1 1
1 2 2
2 2 3


4) Then I built a transformation matrix by finding the coordinates of e1, e2, and e3 relative to v1, v2 and v3

I got
1 -1 0
0 1 -1
0 0 1

I calculated an inverse

1 1 0
0 1 1
0 0 1

5) Then if I had done everything right
[T]' should have been equal to Q[T]Q-1

But it wasn't even close

1 1 1
0 1 1
0 0 1

Any help would be greatly appreciated.
Margaret
 
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  • #2
3) Then I think is wrong but I'm not sure

I built another matrix [T]' = T(v1) | T(v2) | t(v3)
where v1, v2, and v3 are the the vectors given above

I got
1 1 1
1 2 2
2 2 3


One error is that T(v1) should be (1,1,1)t instead of (1,1,2)t. A more important error is that the resulting numbers "1,1,1" are coefficients with respect to the standard basis, not B'. You need to figure out x, y, and z such that
(1,1,1)t=x v1 + y v2 + z v3. This is easy to do by inspection: x=0, y=0, z=1. The first column of [T]' is therefore (0,0,1)t. Do a similar computation for T(v2) and T(v3). The appropriate coefficients x, y, and z are easy to find by inspection, although not as trivial as for T(v1).

Also, the inverse of

1 -1 0
0 1 -1
0 0 1

is almost but not quite

1 1 0
0 1 1
0 0 1

The row 1 column 3 entry is incorrect.
 
  • #3
Thank you very much. I think I've got it now.
 

1. What is a Matrix of Transformation?

A Matrix of Transformation is a mathematical representation of a linear transformation between vector spaces. It is a rectangular array of numbers or symbols arranged in rows and columns that can be used to perform operations on vectors.

2. What is a non standard basis?

A non standard basis is a set of vectors that are not the standard basis vectors (i, j, k) in three-dimensional space. These vectors can be used to represent a different coordinate system or to describe a transformation in a non-standard way.

3. How is a Matrix of Transformation related to a non standard basis?

A Matrix of Transformation is used to represent a linear transformation between vector spaces, and it can be computed using the non standard basis vectors. The column vectors in the matrix represent the images of the basis vectors under the transformation.

4. What is the purpose of using a non standard basis?

Using a non standard basis can simplify the representation of a transformation and make it easier to perform calculations. It can also provide a different perspective or understanding of the transformation by using a different set of basis vectors.

5. How do you convert a Matrix of Transformation to a standard basis?

To convert a Matrix of Transformation to a standard basis, you can use the change of basis formula. This involves finding the inverse of the matrix representing the non standard basis and multiplying it by the Matrix of Transformation. The resulting matrix will be in terms of the standard basis vectors.

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