Let V be a real inner product space. Prove that for all 𝐯, 𝐰 in V (a) 𝐯+𝐰^2-𝐯-𝐰^2=4⟨𝐯, 𝐰⟩(b) 𝐯+𝐰

step 1 So first we need to prove that, I'm looking for my paper here. Okay, this says v plus w squared

Let V be a real inner product space. Prove that for all 𝐯, 𝐰...

Key Concepts

Real Inner Product Space

A real inner product space is a vector space over the real numbers provided with an inner product, which is a function that attaches a real number to each pair of vectors. This structure not only allows the definition of angles and lengths, but also gives rise to geometric interpretations similar to those in Euclidean spaces.

Norm Induced by Inner Product

The norm of a vector in an inner product space is defined in terms of the inner product, typically as the square root of the inner product of the vector with itself. This norm measures the 'length' of a vector and is essential for understanding geometric properties and expressions related to distances and angles.

Bilinearity and Symmetry of the Inner Product

The inner product in a real vector space is bilinear and symmetric, meaning it is linear in each argument and its value does not change when the order of the vectors is switched. These properties allow one to expand expressions involving inner products of sums or differences of vectors, leading to convenient algebraic manipulations.

Parallelogram Law

The parallelogram law is a fundamental identity in inner product spaces that relates the sum of the squares of the norms of the sum and difference of two vectors to the sum of the squares of the norms of the vectors themselves. This law is pivotal in deriving many identities and understanding the geometric structure implied by an inner product.

Transcript

00:01 So first we need to prove that, i'm looking for my paper here.

00:18 Okay, this says v plus w squared minus v minus w squared equals 4v2.

00:42 Interesting.

00:43 So the norm of v plus w squared would be, let me see here, it would be the inner product of v, these are all vectors, plus w and v plus w.

01:30 And of v.

01:32 Okay.

01:32 And of course the norm.

01:35 Of v minus w squared is going to be v minus w and v minus w.

01:53 So let's write this out.

01:55 The inner product of v plus w and v plus w minus w minus minus minus minus the inner product of v plus w minus the inner product of v minus w and v minus w.

02:21 That's just using the definition of the norm.

02:25 Now, according to property 4, we can make this say v, v plus w plus w plus w, v plus w, v plus w, v plus w, v.

03:08 Whoops.

03:09 Got my symbols in the wrong place.

03:21 Okay, so v, v minus w minus negative w.

03:43 V minus w.

03:44 V minus w.

03:50 Which, of course, we could using property three factor out the negative one.

04:00 So that gives us positive.

04:03 Okay, using property two, we can switch these around and now, using property four again, we can separate them.

04:47 V, v, plus wv, plus v, w, minus v, v, minus negative w, v, v plus, oh, i'm running out of room, i'm gonna have to go to the next line.

05:44 Vw plus negative w, w.

05:51 And then according to property three, we can put the negative outside here, and we can change these two.

06:00 Positive.

06:02 Okay, vv minus vv.

06:08 Ww minus ww.

06:14 And i'm going to use property two to rewrite all of these as vw.

06:22 So it's vw, the inner product of vnw, plus the inner product of vnw, plus the inner product of vnw, plus the inner product of vnw, plus the inner product of of v and w plus the inner product of v and w.

06:44 And there are four of them.

06:52 And that's what we intended to prove.

06:55 B.

06:59 Now we need to show that the norm of v plus w squared plus v minus w squared, well that's the same thing.

07:26 V plus w squared, v minus w squared, is 2 times the norm of v squared plus the norm of w squared.

07:45 Okay, so this is actually the same.

07:48 Oh no, we got a plus in between...

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