Answer check and explanation(Linear transformation)

[negation]

Homework Statement

Find the standard matrix of the following linear transformation:

T(x₁, x₂, x₃, x₄) = (-2 x₁ - 5 x₂ - 4 x₃ - x₄, 2 x₁ + 2 x₂ - 5 x₃ + x₄)

The Attempt at a Solution

[x₁,x₂,x₃,x₄] [-2,2;-5,2;-4,-5;-1,1]
=[-2 x₁ - 5 x₂ - 4 x₃ - x₄, 2 x₁ + 2 x₂ - 5 x₃ + x₄]



T(e₁) = (-2,2)
T(e₂) = (-5,2)
T(e₃) = (-4,-5)
T(e₄) = (-1,1)

A = [-2,-5,-4,-1; 2,2,-5,1]

The answer has been checked to be correct. But I'm not seeing why the standard matrix has to be transposed?

 

[LCKurtz]

A matrix representing a transformation from $R^4 \to R^2$ would be a 2x4 matrix. The images of the basis vectors give the columns. Nothing is transposed that I see.

 

[Mark44]

Homework Statement

Find the standard matrix of the following linear transformation:

T(x₁, x₂, x₃, x₄) = (-2 x₁ - 5 x₂ - 4 x₃ - x₄, 2 x₁ + 2 x₂ - 5 x₃ + x₄)

The Attempt at a Solution

[x₁,x₂,x₃,x₄] [-2,2;-5,2;-4,-5;-1,1]
=[-2 x₁ - 5 x₂ - 4 x₃ - x₄, 2 x₁ + 2 x₂ - 5 x₃ + x₄]

There are two problems with the above:
1. You have your x vector on the wrong side of A (it should be Ax rather than xA), and your matrix is wrong. You have four rows with two columns - it should be two rows with four columns.

T(e₁) = (-2,2)
T(e₂) = (-5,2)
T(e₃) = (-4,-5)
T(e₄) = (-1,1)

The vectors on the right are all column vectors.

A = [-2,-5,-4,-1; 2,2,-5,1]

The answer has been checked to be correct. But I'm not seeing why the standard matrix has to be transposed?

The transformation is T: R⁴ → R², so the matrix for T will by 2 X 4 (i.e., two rows with four columns each).

 

[negation]

There are two problems with the above:
1. You have your x vector on the wrong side of A (it should be Ax rather than xA), and your matrix is wrong. You have four rows with two columns - it should be two rows with four columns.
The vectors on the right are all column vectors.
The transformation is T: R⁴ → R², so the matrix for T will by 2 X 4 (i.e., two rows with four columns each).

Both of you guys are correct.