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Suppose $\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}$ and $\mathbf{v}_{1}, \ldots, \mathbf{v}_{q}$ are vectors in a vector space $V,$ and let$H=\operatorname{Span}\left\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}\right\}$ and $K=\operatorname{Span}\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{q}\right\}$Show that $H+K=\operatorname{Span}\left\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}, \mathbf{v}_{1}, \ldots, \mathbf{v}_{q}\right\}$

$H+K=\operatorname{Span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{p}, \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{q}\right\}$

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

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in this video game is selling problem number 34 of section 4.1 Ah, which basically gives us a scenario where you want to u P and G Want to repeat our vic? You are vectors and a vector space of E and H equal span. You want through U P and K Eagle span of the one true week you and we need to show that age Plus, K is this is the span of the, uh is the span from you. One is a span of you want through U P and G 12 week you So the spans air just combine. So this is pretty simple. So any vector of the form h plus que there any, um, any vector the off h plus que of the set h Miss Kay would be in the form of you plus V where you is in h and V is in K because we're just adding the to, uh, vector spaces. And since since you is an h, you know that any, um, vector from that subspace would have to be a linear combination of the span about subspace. So you know that any vector you in h would be, um, some scaler x one times you won plus, um x two times you two, which goes on toward X and x p times u p and the same is true with V and K were you will have will also have to be a linear combination of factors were, um, RV as no is the linear combination of some killer a one plus, um, be one plus a two times of you, too, Plus all the way thio a que Times v Q. So since we have these individual Indian combinations can combine them and you possibly where it's x one u one plus all the way to x p u p plus a one b one. It was all the way, Thio. Thank you, b Q. So since both these vectors air individually of inner combination and H plus que just equals miss linear combination, Um, by the definition of span, we know that h plus que is spend you won. There were U P and everyone through thank you. Because the vectors individually and there's all spaces are, um, some linear combinations of the vectors and that some sweeties So if you just combine them, it would give us, um the answer

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