Question

Describe the possible echelon forms of the standard matrix for a linear transformation T where​ T: set of real numbers R cubedright arrowset of real numbers R Superscript 4 is​ one-to-one.Give some examples of the echelon forms. The leading​ entries, denoted box​, may have any nonzero value. The starred​ entries, denoted star​, may have any value​ (including zero). Select all that apply.

Answer 1

Answer:

║Y *  * ║

║0 Y * ║

║0 0 Y║

║0 0 0║

Step-by-step explanation:

First, let's look at the information in the question:

The proof of the theorem proves that the columns, say, A and B must be independent.

In the matrix, there is linearity and the trivial solution would be [ 0, 0 , 0 , 0] for the matrix to exist. In other words, all the unknowns must be zero.

This establishes the fact that the matrix T needs to be independent for the matrix function, say T (x) to be one-to-one.

By definition, one-to-one is ker (T) = {0}

Thus, the null space is occupied only by the zero vector.

Note: if there was no linear independence in the vectors, the solution would not be zero in the vector.