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If $L$ is $n \times n$ and the equation $L \mathbf{x}=\mathbf{0}$ has the trivial solution, do the columns of $L$ span $\mathbb{R}^{n} ?$ Why?

since the key word only is missing from the given statement the columns of L will not always span $R^{n}$ .

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

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Baylor University

University of Michigan - Ann Arbor

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in this problem we have and l, which is a, uh, and by end square matrix. And we also have the senior system l ax equal Cyril. And we want to know that whether the columns of l spend our in So the answer is no. Because, um, you know, considered the linear system l X equals zero for this system. This equation this system has attributes. Um, every system like this has a trivial solution, and and therefore, l may not be a convertible. Beatrix and the columns off l there will not, probably would not suspend. Ah, the whole space are in here. So to see, then to see an example, um, that's check has led. Um, the Matrix l to be, uh, maybe a three by three matrix with all zeros. You're zero and 000 Okay, so this system clearly has has a trivial solution, but the span you check the span, it's all in the span, is formed up three zero vectors. So the span is three zero vectors. Right? So it's very clear that these 30 vectors will not spend the whole space of our three. And this man does not spend does not, It's been are three. So that's the example that you my interested because this this shows that it is, um this is called him Something off the matrix l does not. Does not have to spend are in

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