Suppose 𝐯1, …, 𝐯m span V. Let 𝐯m+1, …, 𝐯n ∈V be any other elements. Prove that the combined coll

Suppose 𝐯1, …, 𝐯m span V. Let 𝐯m+1, …, 𝐯n ∈V be any other...

Key Concepts

Redundancy in Spanning Sets

Redundancy in spanning sets refers to the idea that adding additional vectors to a spanning set does not change the fact that the set spans the vector space. Even if the extra vectors are not necessary to span the space, the expanded set still spans the space because the original spanning set is contained within it.

Linear Combination

A linear combination involves forming a new vector by multiplying given vectors by scalars and then adding the results. This concept is used to express the condition for a set to be spanning: every vector in the vector space must be representable as a linear combination of the vectors in the spanning set.

Vector Space

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars while satisfying specific axioms. Understanding vector spaces is essential because spanning sets, linear combinations, and many other fundamental concepts in linear algebra are defined within the context of vector spaces.

Spanning Set

A spanning set for a vector space is a collection of vectors such that every element of the vector space can be expressed as a linear combination of these vectors. This concept is central to the problem because proving that a collection spans the vector space involves demonstrating that any vector in the space can be formed using the vectors from the set.

Transcript

00:01 So we have a spanning set for some vector space, v1 through vn, and we'd like to show that if we do this to it, just subtract each subsequent vector from each vector, except the last one which stays the same, then it is still a spanning set.

00:16 So we can do that here.

00:17 Let's let v be an arbitrary vector in v, so we can write v as a sum of vi, so let's write that here, sum alpha i times vi, as i ranges from 1 to n, for some scalars alpha i.

00:34 We would like to show that we can also find sum i equals 1 to n beta i of, let's see, vi minus vi plus 1.

00:48 I suppose this should only be from 1 up to n minus 1, because then we want to add to that beta n times vn, because that one has not been changed.

01:05 To this end, actually, all i'm going to do is expand this out.

01:10 We want to find that this is the case.

01:14 We don't know that it's true yet, but if we push some symbols around here a little bit, this guy is equal to the sum from i equals 1 to n minus 1 beta i vi, plus the sum from i equals 1 to n minus 1 of beta i vi plus 1, minus i should say, plus beta n vn.

01:45 That's fine.

01:47 We can actually push this around a little bit.

01:51 So i equals, this is going from 2 to n, and we can get beta i minus 1 here, beta i minus 1 of vi, not vn, which we can actually then write this out a little bit, combine things here, we'll have beta 1 v1 plus the sum as i goes from 2 to n minus 1, beta i minus 1 vi, oh, excuse me, this will be beta i minus beta i minus 1 vi, plus beta n vn.

02:49 So all i've done here is push some stuff around using linearity.

02:52 We have some scalar times vi minus vi plus 1 is what each of these are, with vn on its own, split that into two sums, and then i've combined the two sums partially into all of this.

03:11 But, what i'm going to claim here is that this looks like if we set gamma 1 to be equal to beta 1, gamma i to be equal to beta i minus beta i minus 1, for i going from between 2 and n minus 1, for 2 less than or equal to i less than or equal to n minus 1, and gamma n equal to beta n, then this guy is just the sum as i goes from 1 to n of gamma i times vi.

03:55 We have something times v1, something times v2, v3, v4, up to vn minus 1, and something times vn.

04:05 So we can manipulate something that looks like this, a linear combination of these, into a linear combination of just the vi.

04:16 What that in particular means is that if we can go from, here we've gone from the betas to the alphas, if we can go from the alphas to figure out what the betas should be, then we'll be able to express any vector v as a linear combination of these vectors.

04:35 Let's try and do that.

04:37 We want now, our goal, is to define the beta coefficients such that alpha 1 equals beta 1, alpha n equals beta n, and alpha i equals beta i minus beta i minus 1, for 2 is less than or equal to i, is less than or equal to n minus 1...

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