Let {v1,v2,v3} be a basis of a vector space V over R let T:V-V be the linear transformation determined by Tv1=V1,Tv2=V2-V3 Tv3=V2 V3 find the matrix of transformation T with {V1 V2,V2-V2,V3} as a basis of both domain and codomain?

Question

Answer

Solution:

Given:
Basis of vector space V over R = {v1,v2,v3}
Linear transformation T: V-V
Tv1=V1, Tv2=V2-V3, Tv3=V2
Basis of domain and codomain = {V1 V2,V2-V2,V3}

To find:
Matrix of transformation T

Approach:
To find the matrix of transformation T with respect to the given basis, we need to express the images of the basis vectors in terms of the basis vectors of the codomain. Then we can form a matrix whose columns are the coordinate vectors of the images of the basis vectors.

Let's begin by finding the coordinate vectors of the images of the basis vectors with respect to the given basis of the codomain.

Coordinate vectors of the images of the basis vectors:

[Tv1] = [V1] (already given)
[Tv2] = [V2-V3] = [V2] - [V3] = [0,1,-1]
[Tv3] = [V2] = [0,1,0]

Now we can form the matrix of transformation T with respect to the given basis of the domain and codomain.

Matrix of transformation T:

| 1 0 0 |
| 0 1 -1 |
| 0 1 0 |

Explanation:

- The first column of the matrix represents the coordinates of the image of v1 with respect to the given basis of the codomain. Since Tv1=V1, the first column is [1,0,0].
- The second column of the matrix represents the coordinates of the image of v2 with respect to the given basis of the codomain. Since Tv2=V2-V3, the second column is [0,1,-1].
- The third column of the matrix represents the coordinates of the image of v3 with respect to the given basis of the codomain. Since Tv3=V2, the third column is [0,1,0].

Note: The matrix of transformation T is also called the change of basis matrix from {v1,v2,v3} to {V1,V2,V3}.

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