Question

Suppose T: ℝ² → ℝ³ is a linear transformation. Let u and v be the vectors given below, and suppose that T(u) and T(v) are as shown:

T(u) = [ 2, 0, -3 ; 1, 1, -3 ]
T(v) = [ -1, 1, 0 ; 1, 1, 0 ]

Find the standard matrix of the linear transformation T.

Answer 1

The question requests to find the standard matrix of a linear transformation T from ℝ² to ℝ³. The standard matrix is constructed by placing images of the standard basis vectors into the matrix columns. However, the provided T(u) and T(v) are not formatted correctly as three-dimensional vectors, and therefore, the task cannot be completed without the correct information.

Step-by-step explanation:

The question involves finding the standard matrix of a linear transformation T from ℝ² to ℝ³. We can construct this matrix by placing the images of the standard basis vectors of ℝ² into the columns of a matrix. The vectors u and v given are the images of the standard basis vectors under the linear transformation T. To find the standard matrix A, we place T(u) as the first column and T(v) as the second column of the matrix.

However, the provided information seems to be a set of typos and not the images of u and v under T. It does not follow the format of a proper linear transformation image from ℝ² to ℝ³. Ideally, T(u) and T(v) should each be three-dimensional vectors, not sets of vectors or matrices. Therefore, we cannot provide the standard matrix without the correct images of the standard basis vectors under T.