A) To find the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain, we need to express the images of the basis vectors under T in terms of the basis vectors.

Given that T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3, we can write these as column vectors to form the matrix:

T = [2 1 0]
[0 2 1]
[0 0 2]

This is the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain.

B) To find T(e2), we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in terms of the basis B, this is [1, 2, 0]^T.

C) To find T(e2) in the standard basis, we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in the standard basis, this is [1, 2, 0]^T.

Question

A) To find the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain, we need to express the images of the basis vectors under T in terms of the basis vectors.

Given that T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3, we can write these as column vectors to form the matrix:

T = [2 1 0]
    [0 2 1]
    [0 0 2]

This is the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain.

B) To find T(e2), we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in terms of the basis B, this is [1, 2, 0]^T.

C) To find T(e2) in the standard basis, we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in the standard basis, this is [1, 2, 0]^T.

Knowee AI · Accepted Answer

A) To find the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain, we need to express the images of the basis vectors under T in terms of the basis vectors.

Given that T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3, we can write these as column vectors to form the matrix:

T = [2 1 0]
    [0 2 1]
    [0 0 2]

This is the matrix of the linear map T with respect to the basis B in the domain and the basis B in the co-domain.

B) To find T(e2), we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in terms of the basis B, this is [1, 2, 0]^T.

C) To find T(e2) in the standard basis, we need to apply the transformation T to the second standard basis vector e2. In this case, e2 corresponds to v2 in our basis B. So, T(e2) = T(v2) = v1 + 2v2.

Expressed as a column vector in the standard basis, this is [1, 2, 0]^T.

[5marks] letT:C3→C3bealinearmapsuchthat T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3. A) Find the matrix 􏰓T􏰔 , i.e., the matrix of the linear map T with respect to B,B the basis B in the domain and the basis B in the co-domain. B) Find 􏰋T􏰀e2􏰁􏰌 . B C) Find T􏰀e2􏰁.

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