In R^n show that (a) 2𝐱^2+2𝐲^2=𝐱+𝐲^2+𝐱-𝐲^2 (This is known as the parallelogram law.) (b) 𝐱-𝐲𝐱+𝐲

In R^n show that (a) 2𝐱^2+2𝐲^2=𝐱+𝐲^2+𝐱-𝐲^2 (This is known as...

In $\mathrm{R}^{n}$ show that (a) $2\|\mathbf{x}\|^{2}+2\|\mathbf{y}\|^{2}=\|\mathbf{x}+\mathbf{y}\|^{2}+\|\mathbf{x}-\mathbf{y}\|^{2}$ (This is known as the parallelogram law.) (b) $\|\mathbf{x}-\mathbf{y}\|\|\mathbf{x}+\mathbf{y}\| \leq\|\mathbf{x}\|^{2}+\|\mathbf{y}\|^{2}$ (c) $4\langle\mathbf{x}, \mathbf{y}\rangle=\|\mathbf{x}+\mathbf{y}\|^{2}-\|\mathbf{x}-\mathbf{y}\|^{2}$ (This is called the polarization identity.)

In $\mathrm{R}^{n}$ show that
(a) $2\|\mathbf{x}\|^{2}+2\|\mathbf{y}\|^{2}=\|\mathbf{x}+\mathbf{y}\|^{2}+\|\mathbf{x}-\mathbf{y}\|^{2}$ (This is
known as the parallelogram law.)
(b) $\|\mathbf{x}-\mathbf{y}\|\|\mathbf{x}+\mathbf{y}\| \leq\|\mathbf{x}\|^{2}+\|\mathbf{y}\|^{2}$
(c) $4\langle\mathbf{x}, \mathbf{y}\rangle=\|\mathbf{x}+\mathbf{y}\|^{2}-\|\mathbf{x}-\mathbf{y}\|^{2}$ (This is called the polarization identity.)

Key Concepts

Euclidean Norm

The Euclidean norm assigns a non-negative length or size to each vector in ?? and is defined in terms of the inner product. It encapsulates the idea of measuring distance or magnitude in a geometric space, obeying properties such as homogeneity, positivity, and the triangle inequality. This concept is fundamental for establishing relationships between vectors and for analyzing geometric structures in vector spaces.

Inner Product

An inner product is a function that takes two vectors and produces a scalar, often encapsulating the notions of angle and length. In ??, this is commonly realized as the dot product, which enables the definition of orthogonality, projections, and notions of

Transcript

00:01 In this problem, we're asked to show a few relationships, inequalities, or inequalities, between norms and inner products.

00:09 So to start with, let's look at the first one.

00:13 On the right -hand side of the first one, we have the norm of x plus y squared plus the norm of x minus y squared.

00:24 And if we write this as dot products, this is just equal to x plus y dotted with x plus y plus y plus x minus y dotted with x minus y and now we can actually foil each of these to get x dotted with x plus two times x dotted with y plus y dotted with y that's the first one here and when we do the same thing with the second one we get x dotted with x minus two times x dotted with y plus y dotted with y and of course these two terms cancel plus 2 x dotted with y minus 2 x dotted with y we end up with 2 x dotted with x plus 2 y dotted with y and this is just 2 times the squared norm of x plus 2 times the squared norm of y which is what we were hoping to show for the next one we can actually think about the quantity x minus y dotted with x minus y times and i'll use brackets to denote that this is a dot product times x plus y dotted with x plus y and this is going to be equal to again remember that we're multiplying the two brackets x dotted with x minus two x dotted with y plus y dotted with y times x dotted with x plus two times x dotted with y plus y dotted with y plus y dotted with y now notice that we can rewrite these as the squared norm of x plus the squared norm of y minus 2x dotted with y times the squared norm of x plus the squared norm of x plus 2x dotted with y.

02:26 And if we notice that we have some quantity minus another quantity times some quantity plus that quantity, this means that we end up with x norm squared plus y norm squared squared minus 2 times x dotted with y squared.

02:45 Now notice we have something interesting.

02:47 This quantity on the left here is positive and this quantity is positive and this quantity is positive...

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