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Suppose $A$ is an $m \times n$ matrix with the property that for all $\mathbf{b}$ in $\mathbb{R}^{m}$ the equation $A \mathbf{x}=\mathbf{b}$ has at most one solution. Use the definition of linear independence to explain why the columns of $A$ must be linearly independent.

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

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in this video, we're starting off by assuming that the Matrix equation a X equals B will have at most one solution for all vectors, be that we might choose. Let's take some logical steps to see where we can get from this statement. Well, if this works where we have at most one solution for all be, let's let be be equal to the zero vector. It should work as well. For the zero vector by the statement, then a X equals zero vector has at most or at most is probably the most important key word here. One solution. So we know X equals zero vector is going to be consistent and we're only going to get to one solution. But we already know that the Vector X equals zero vector is the trivial solution to this equation. So because we have a homogeneous equation with those euro vector here, placing as euro vector in this position gives us a solution which we call the trivial solution. But that means we have found one solution and there is only one solution because of the word at most. So let's say what that means Next were right to therefore a X equals zero vector has on Lee the trivial solution thanks to the at most statement here again, probably the most important statement in this entire problem. No, when we say X equals of zero Vector has on Lee the trivial solution that gives us another conclusion immediately about this matrix A. We have now that the columns of a are linearly independent. So we're able to say a very strong statement about the columns of a really, just from this information, which is quite surprising.

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