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Explain why the columns of an $n \times n$ matrix $A$ are linearly independent when $A$ is invertible.

linearly independent

Algebra

Chapter 2

Matrix Algebra

Section 2

The Inverse of a Matrix

Introduction to Matrices

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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So the metrics, eh, is our n by N Matrix and we right this way when we just highlight the columns. So C one is the first column. CTO is the second column and so forth up to CNN, which is the andthe column where the ice call him. He's just a column vector. See, I won C i n in our in now. If he right the product, a times a vector x one xn and we expand this product, we see that these equals just x one times the first column, plus x two times the second column, plus all the way to exciting times. The andthe column and these expression here is a vector in a are and and more specifically in the span off the vectors. See one CNN. So now the span of C one CNN is equal toe rn, by definition, even leave every vector me off rn is in the span of C one CNN, but we have served there. We can write on a genetic element of the spawn off. See Juan Si n. It's just the metric say that multiplies these nomadic, efficient ce X one exxon. So he's is equivalent to saying that the system eh x one xn equals the vector V has solutions the next one accent for every vector V in Oran. But now we know that the system a solutions for every vector we if and only if the medics say is in vertebral as we wanted to see.

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