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Use the inner product axioms and other results of this section to verify the statements in Exercises $15-18$ .$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 7

Inner Product Spaces

Vectors

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

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In mathematics, a vector (…

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Use the inner product axio…

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Prove from the inner produ…

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Use the previous exercise …

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Let $V$ be a real inner pr…

01:44

Let $\mathbf{u}=\left\lang…

04:02

PROOF Prove the following.…

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Prove the theorem. Use the…

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Prove the following. $…

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Given $u=\langle a, b\rang…

let's combine in just one computation, both the plus and minus case. So the length of you plus or minus V squared. So this is you plus or a minus V against you, plus or minus B. But with these notation, we mean that we either always take the blasts. All we always take the miners now is the linearity in the first argument that's you against you. Plus or minus V plus or minus V against you, plus or minus B. Now we're still in a charity. In the second argument, we got you against you plus or minus you against V Blaster miners. And now we put in brackets the whole second in their product. So we have V against you blossom minus V against me. Now observe that you against you is just the length of your squared. And then we have plus reminders twice you ve where the two comes from. The fact that both you and we and we and you are the same because of the symmetry. And we're something them with the same sign. So it's other two pluses or two minuses. So it's plus or minus two you ve and then the last term is a plus in green because he re multiplying pastor minors by plus or minus so either that both plus or the boat minus. When we multiply to turn together, you always get plus, So the remaining term is plus V against Visa that the length of B squared an hour done. We just write the sum. And so the plus reminder simplifies out zero, and you have twice the length of you squared, plus twice the length of the square as we wanted.

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