Let A and U be as shown below, and define a transformation T:ℝ3 → ℝ2 by T(V) = AV. Find T(U).
A = −1 0 −1
−5 0 3
U = −1 −5 1
To find T(U), we need to find the image of the vector U under the transformation T.
Given that T(V) = AV, where V is a vector in ℝ^3, we can find T(U) by multiplying the matrix A by the vector U.
To do this, let's first write out the matrix A:
A = -1 0 -1
-5 0 3
And the vector U:
U = -1
-5
1
Now, to compute T(U), we multiply the matrix A by the vector U using matrix multiplication:
T(U) = A * U
= -1 0 -1 * -1
-5 0 3 -5
1
To perform the matrix multiplication, we multiply corresponding elements and sum them up:
T(U) = (-1 * -1) + (0 * -5) + (-1 * 1)
(-5 * -1) + (0 * -5) + (3 * 1)
= 1 + 0 - 1
5 + 0 + 3
= 0
8
Therefore, T(U) = <0, 8>.