Question

Finding matrix representing transformation relatibe to a basis

Answer 1

Answer:

Let T be the linear transformation from the vector space R2 to R2 itself given by

T([x1x2])=[3x1+x2x1+3x2].

(a) Verify that the vectors

v1=[1−1] and v2=[11]

are eigenvectors of the linear transformation T, and conclude that B={v1,v2} is a basis of R2 consisting of eigenvectors.

(b) Find the matrix of T with respect to the basis B={v1,v2}.

Add to solve later

Contents [hide]

Problem 314

Solution.

(a) B={v1,v2} is a basis of R2 consisting of eigenvectors

(b) Find the matrix of T with respect to the basis B

Solution.

(a) B={v1,v2} is a basis of R2 consisting of eigenvectors

We compute that

T(v1)=T([1−1])=[2−2]=2[1−1]=2v1

and

T(v2)=T([11])=[44]=4[11]=4v2.

Thus, v1 is an eigenvector corresponding to the eigenvalue 2 and v2 is an eigenvector corresponding to the eigenvalue 4.

Since v1,v2 are eigenvectors corresponding to distinct eigenvalues, they are linearly independent, and thus B={v1,v2} is a basis of R2.

(b) Find the matrix of T with respect to the basis B

From the computation in part (a), we have

T(v1)=2v1+0v2T(v2)=0v1+4v2.

Hence the coordinate vectors of T(v1),T(v2) with respect to the basis B={v1,v2} is a basis of R2 are

[T(v1)]B=[20],[T(v2)]B=[04].

Thus the matrix A of the linear transformation T with respect to the basis B is

A=[[T(v1)]B,[T(v2)]B]=[2004].