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Let $S$ be a finite set in a vector space $V$ with the property that every $x$ in $V$ has a unique representation as a linear combination of elements of $S .$ Show that $S$ is a basis of $V .$

Thus, the property (ii) for that $S$ is a basis for $V$ is proved.since the set $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\}$ is a spanning set for $V$ and the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}$ arelinearly independent, it forms basis for $V$ .

Calculus 3

Chapter 4

Vector Spaces

Section 4

Coordinate Systems

Vectors

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Okay, So our question is, let s be a finance set. Okay? Finance it in a victory space victory space. We with the property that every X in V has a unique representation. Okay. Unique representation as a linear combination off elements off. Yes. So we have to show that, as is a basis off we Okay, so we know that we know that as is finite dimensional. Okay, is finite dimensional. So we can say that this is it. And dimensional ticket and dimensional. So on this elements are as I and for their values, less than it was toe end and greater than it was 21 Okay. And for that on bet Visa victory space. So this will be fine. So each each eggs and we has a unique representation. Okay? Unique representation as a linear combination off as a linear combination off combination off s. And as to s three. And so on s and Okay, so where definition as an as his spending set for V okay. And in particular, that is also means that zero has a unique representation. Okay, I tried to understand that Nick represent representation, okay? And currently we cancer. That this killer C one C two C three and so on. C and the victory questions. Okay. In the world of the factory question, it will be C one s one plus he two as two plus C three s three and so on. C n s m equals to zero. You know that from this. So this will have a unique solution. And the travel unique solution. And the solution is Jiro on zero. It means thistles, the travel solution. And we know travel solution is always the solution. Okay? And it is the only solution. So it means, by definition, it means as one as two s and so on s and is linearly dependent. Okay, because it has travel solution and only solution. So it is linearly independent. Okay, So any as Iselin independent spending set that does, by definition, definition, as is a basis off movie. Okay? And we can say that as it was to s equals two. We want we do. We three we and you can save as one as two s three s and is a spending set off. We okay? And these are linearly dependent. So the definition as he is a basis off we it is prude. Thank you.

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