Find the matrix representation of the linear transformation T : R3 → R2 givenby the mapxyz 7 →( x + yx + z), where B =111 ,110 ,100and D ={( 10),( 02)}are bases for R3 and R2 respectively
Question
Find the matrix representation of the linear transformation T : R3 → R2 givenby the mapxyz 7 →( x + yx + z), where B =111 ,110 ,100and D ={( 10),( 02)}are bases for R3 and R2 respectively
Solution
To find the matrix representation of the linear transformation T : R3 → R2, we need to apply the transformation to each of the basis vectors in B and express the result in terms of the basis vectors in D.
- Apply T to the first basis vector in B:
T(1,1,1) = (1+1, 1+1) = (2,2)
Express this in terms of the basis vectors in D:
2*(1,0) + 2*(0,2) = (2,2)
So the first column of the matrix representation is (2,2).
- Apply T to the second basis vector in B:
T(1,1,0) = (1+1, 1+0) = (2,1)
Express this in terms of the basis vectors in D:
2*(1,0) + 1*(0,2) = (2,1)
So the second column of the matrix representation is (2,1).
- Apply T to the third basis vector in B:
T(1,0,0) = (1+0, 1+0) = (1,1)
Express this in terms of the basis vectors in D:
1*(1,0) + 1*(0,2) = (1,1)
So the third column of the matrix representation is (1,1).
Therefore, the matrix representation of T with respect to the bases B and D is:
| 2 2 1 | | 2 1 1 |
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