There are two problems with the above:negation said: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]
The vectors on the right are all column vectors.negation said:T(e1) = (-2,2)
T(e2) = (-5,2)
T(e3) = (-4,-5)
T(e4) = (-1,1)
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 said: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?
Mark44 said: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).
A linear transformation is a function that maps one vector space to another in a way that preserves the structure of the original vector space. This means that the transformation must satisfy two properties: 1) it must preserve addition and scalar multiplication, and 2) the transformation must send the zero vector to the zero vector.
To check if a transformation is linear, you can perform two tests: 1) the addition test, where you check if the transformation preserves addition, and 2) the scalar multiplication test, where you check if the transformation preserves scalar multiplication. If the transformation passes both tests, it is considered linear.
The matrix representation of a linear transformation is a way to represent the transformation using a matrix. This is done by choosing a basis for the original vector space and a basis for the target vector space. The columns of the matrix are then the images of the basis vectors of the original vector space under the transformation.
The dimension of the target vector space after a linear transformation is equal to the number of linearly independent columns in the matrix representation of the transformation. This is known as the rank of the transformation.
Linear transformations are important in science because they allow us to model and understand relationships between variables in a linear way. This is useful in various fields such as physics, engineering, and economics, where many real-world phenomena can be described and analyzed using linear transformations.