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Let $S$ be a subset of an $n$ -dimensional vector space $V,$ and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V .$

the subset S cannot span $\mathrm{V}$ .

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

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for this example. We start out with age. It's going to be a vector subspace of some vector space V, and then we're going to make an assumption that we have two vectors U and V, which are also in H. So what we're going to do in this situation is verified the following claims. We claim that the span of those two vectors U and V, is contained in H. I'll use a symbol subset to say that the span of UV is a subset of age, which is a cinnamon synonym of saying it's contained. Let's see how to prove this. First, we begin by taking an element. We could call it X if we like that comes from the span. So let's say let X be in the span of the vectors you envy. Our goal here is to show that if X came from the span, then it must be in the set. H if we can show that that our work is done and we would have shown containment. So let's go on to the next step. Then, as soon as we know that the Vector X came from this man, that means next that there exists scale er's I'll call them C and D such that X is equal to see Time's You plus d times V. The reason we could write that such an equation like this is FX came from the span of U and V. That just means it's a linear combination of you envy. And this is one such linear combination now notice. So far we have nowhere used the fact that H is a vector subspace. So let's begin doing that next. We know that since H is a subspace of the and we know that U and V came from H, which is also critically important, it follows that See, you and Devi are in H now why is that true? Well, the reason is, if we have a subspace, a subspace is closed under scaler multiplication. And so this is a scaler multiplication of a vector u a vector v which came from H so they must now both b NH as well. Whatever the closure, do we have? Well, another closure property is if we use vector addition now of two vectors that are known to our day b NH then the some of those vectors must also be in age. I will say likewise X which is equal to see you plus DV is in a church. So this was our goal. This shows that the spin vectors U and V are contained in the subspace H, as required.

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