The discussion centers on finding a matrix representation for the transpose mapping L(A) = A^t with respect to the standard basis in a 4-dimensional space. Participants confirm that the correct matrix representation is a 4x4 matrix, specifically: the first column as [1, 0, 0, 0], the second as [0, 0, 1, 0], the third as [0, 1, 0, 0], and the fourth as [0, 0, 0, 1]. The conversation clarifies that while the standard basis is typically represented as {<1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, <0, 0, 0, 1>}, it is legitimate to work with vectors in ℝ⁴ instead of 2x2 matrices, as they are isomorphic.
PREREQUISITESStudents and educators in linear algebra, mathematicians working with matrix theory, and anyone interested in understanding linear mappings and their representations.
How is [1, 1, 1, 1] a basis?PsychonautQQ said:Homework Statement
Let L be a mapping such that L(A) = A^t, the transpose mapping. Find a matrix representing L with respect to the standard basis [1,1,1,1]
This works, if it's legitimate to work with vectors in ##\mathbb{R}^4## instead of 2 x 2 matrices. Of course ##\mathbb{M}_{2 x 2}## is isomorphic to ##\mathbb{R}^4##. Based on what I think the problem statement is supposed to mean, your solution looks fine.PsychonautQQ said:Homework Equations
The Attempt at a Solution
So should I end up getting a 4x4 matrix here? I got 1,0,0,0 for the first column, 0,0,1,0 for the second column, 0,1,0,0 for the third column and 0,0,0,1 for the fourth column. is this correct?