Question

The columns of the standard matrix for a linear transformation from Rⁿ to Rᵐ are the images of the columns of the n x n identity matrix under T. Choose the correct answer below.

A. True. The columns are directly derived from the identity matrix.
B. False. The columns have no relation to the identity matrix.
C. True. Only square matrices exhibit this property.
D. False. This property is not applicable to standard matrices.

Answer 1

Option (A), The statement is True; the columns of a standard matrix for a linear transformation are the images of the identity matrix's columns under the transformation.

Step-by-step explanation:

The statement 'The columns of the standard matrix for a linear transformation from Rⁿ to R⁽ are the images of the columns of the n x n identity matrix under T' is True. The standard matrix for a linear transformation, T, can be found by applying T to each column vector of the identity matrix in Rⁿ. This results in a set of m-dimensional column vectors that make up the columns of the standard matrix.

A matrix A is said to represent the linear transformation T from Rⁿ to R⁽ if for any vector x in Rⁿ, the product Ax is equal to T(x). Therefore, each column of the identity matrix in Rⁿ is transformed by T into a corresponding column in the matrix A, clearly showing that the columns are directly derived from the identity matrix I.