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If $A$ is an $n \times n$ matrix and the equation $A \mathbf{x}=\mathbf{b}$ has more than one solution for some $\mathbf{b},$ then the transformation $\mathbf{x} \mapsto A \mathbf{x}$ isnot one-to-one. What else can you say about this transformation? Justify your answer.

The transformation $\mathbf{x} \mapsto A \mathbf{x}$ is not invertible, not one-to-one and does not map $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$ .

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

Campbell University

McMaster University

Lectures

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in this example, we're dealing with a square matrix of size N by N, and we know that there is a vector be coming out of our end, for which the Matrix equation A X equals B has infinitely many solutions. What's next? Define a transformation, which will be X maps to eight times X or, if you prefer, in a different notation, this is the same as saying t takes are in into our n By the rule, TF X equals eight times X really is just a matter of notation if you prefer the compact one versus the equivalent. But Mawr Link Lee version. So next let's say what we can learn about this transfer mission X maps to a X. To do that, we're going to also cite the convertible matrix theorem as we go. So let's put it about here. So the things that are most interesting to us when we're considering such a transformation our parts I deals with the linear transformation and let's see which other part looks like. Part F also deals with the linear transformation as well, but our transformation deals with the standard matrix A and so let's also used part a So we can say something interesting about that standard matrix. Now let's go to a portion of the given information here that allows us to access Thean vertebral matrix there. That would be this part where we're told a X equals B has infinitely many solutions. So if we go to Parts D here, this part is a statement that a X equals B zero hope that means what you need. A different one. How about part G that will be the part. Will access this part is false. Let's write that here. So since a X equals B has in fine nightly solutions parts G in the convertible matrix, the're, um is false. This implies that every part of the in vertical matrix theory is now false, so we can determine next that the transformation t or if you prefer, the transformation, X maps to a X is not one 21 that comes from Let's see here Looks like part F. So that's by part F. Let's see what else we could determine here. Well, per I is about the transformation being onto, So we also know that X maps to a X is don't forget the most important word here. Not and on to transformation or I'll use mapping. It's a cinnamon, um, and it fits right here. So we know that the transformation is not 1 to 1. Its not on two. Let's also state part A. Also, a inverse does not exist, and this is by part a. Oh, and the statement above was by part I. So since a inverse does not exist, that also tells us something interesting about this transformation or transformation. T. If you prefer that notation, it tells us X maps to a X because a inverse did not exist is not convertible as a mapping or a function if you prefer. So, knowing that a X equals B has infinitely many solutions tells us many, many things about this transformation. It's not Oneto one, its not on two, and it's not in verbal either.

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