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2. Find the matrix with respect to the canonical basis of the linear transfor- mation T:R+R2 defined by X1 Tx2 (5x, – 2x2 + x3) – + 4x₂ - 7x₂ ха

User Voltan
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4 votes

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 ]

User Z Jones
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