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In Exercises 15 and $16,$ list five vectors in $\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\} .$ For each vector, show the weights on $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ used to generate the vector and list the three entries of the vector. Do not make a sketch.$$\mathbf{v}_{1}=\left[\begin{array}{r}{7} \\ {1} \\ {-6}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{-5} \\ {3} \\ {0}\end{array}\right]$$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 3

Vector Equations

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

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Okay, so we're asked to find five vectors that sit in the span of V one and V two and were asked to give the weights of each factor. So Vector X that sits inside the span will be a linear combination of U and V two. So the simplest ones. So the simplest choices of a one and a two is first, we'll take a one in 82. Equal 00 is always in the span of any vectors, right? Because the because the span is a subspace right in the subspace contains the zero vector, since it is a vector space. And so 000 is one possible answer, and the weights are zero for both of them. The next one is to take this one to be one, and this one to be zero. So 71 minus six sits also inside the span of V one and V two and minus 530 Also spit sentence in the span of the wannabe to the weight on V 10 The weight on V two is one in this case. Then you could take you could just add them. We want to be too So take a one and a two to be both won. And when you do that, you get this. I didn't component by component, or you could do the one minus V two. So the way on V one would be one the way envy to would be minus one. And so you get 12 minus two minus six. And these are five vectors there. Sit in the span. Um, of course, you could choose any anything for a one and a two you like, but this is the simplest one that I could think of.

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