If T is a one-to-one linear transformation from R³ to R⁵, and A is a matrix such that

, then

.

Question

If T is a one-to-one linear transformation from R³ to R⁵, and A is a matrix such that , then .

Quizplus · Accepted Answer

Since T is a one-to-one linear transformation, it preserves linear independence. Therefore, if the columns of matrix A are linearly independent in R³, their images under T will also be linearly independent in R⁵. Thus, the matrix formed by the columns of the image of A under T will also have rank 3, and so its determinant will be non-zero. Therefore, det(T(A)) ≠ 0, and the given statement is true.

If T Is a One-To-One Linear Transformation from R3 to R5

If T Is a One-To-One Linear Transformation from R³ to R⁵