Answer:
The matrix representation of the linear transformation T with respect to the canonical basis of R³ is:
[ 5 0 0 ]
[-2 1 -7 ]
[ 1 4 -7 ]
Explanation:
It seems there might be some formatting issues in your question. To help you with this problem, I'll assume you meant the following linear transformation:
T(x) = (5x₁, -2x₂ + x₃, x₁ + 4x₂ - 7x₃)
where x = (x₁, x₂, x₃) is a vector in R³ (not R² as you mentioned).
Let's find the matrix representation of this linear transformation with respect to the canonical (standard) basis.
The canonical basis of R³ is given by:
e₁ = (1, 0, 0)
e₂ = (0, 1, 0)
e₃ = (0, 0, 1)
We'll compute the image of each basis vector under the transformation T:
1. T(e₁) = T(1, 0, 0) = (5(1), -2(0) + (0), (1) + 4(0) - 7(0)) = (5, 0, 1)
2. T(e₂) = T(0, 1, 0) = (5(0), -2(1) + (0), (0) + 4(1) - 7(0)) = (0, -2, 4)
3. T(e₃) = T(0, 0, 1) = (5(0), -2(0) + (1), (0) + 4(0) - 7(1)) = (0, 1, -7)
Now, form the matrix representation of the linear transformation T by using the images of the basis vectors as columns:
_______
| 5 0 0 |
| 0 -2 1 |
| 1 4 -7 |
So, the matrix representation of T with respect to the canonical basis is:
[ 5 0 0 ]
[-2 1 -7 ]
[ 1 4 -7 ]