Friday, December 2, 2022

# Basis And Span Linear Algebra

## Chapter 2linear Combinations Span And Basis Vectors

Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity.

In the last chapter, along with the ideas of vector addition and scalar multiplication, I described vector coordinates, where theres this back-and-forth between pairs of numbers and two-dimensional vectors.

Now I imagine that vector coordinates were already familiar to many of you, but theres another interesting way to think about these coordinates, which is central to linear algebra. When you have a pair of numbers meant to describe a vector, like , I want you to think of each coordinate as a scalar, meaning think about how each one stretches or squishes vectors.

xy-coordinate system, there are two special vectors. The one pointing to the right with length 1 1, commonly called i hat i ^ \hat i i^ or the unit vector in the x-direction. The other one is pointing straight up with length 1 1, commonly called j hat j ^ \hat j j^ or the unit vector in the y-direction. Now, think of the x-coordinate as a scalar that scales i i^, stretching it by a factor of 3 3, and the y-coordinate as a scalar that scales j j^, flipping it and stretching it by a factor of 2

## Proof That Every Vector Space Has A Basis

Let V be any vector space over some field F. Let X be the set of all linearly independent subsets of V.

The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by â.

Let Y be a subset of X that is totally ordered by â, and let LY be the union of all the elements of Y .

Since is totally ordered, every finite subset of LY is a subset of an element of Y, which is a linearly independent subset of V, and hence LY is linearly independent. Thus LY is an element of X. Therefore, LY is an upper bound for Y in : it is an element of X, that contains every element of Y.

As X is nonempty, and every totally ordered subset of has an upper bound in X, Zorn’s lemma asserts that X has a maximal element. In other words, there exists some element Lmax of X satisfying the condition that whenever Lmax â L for some element L of X, then L = Lmax.

It remains to prove that Lmax is a basis of V. Since Lmax belongs to X, we already know that Lmax is a linearly independent subset of V.

If there were some vector w of V that is not in the span of Lmax, then w would not be an element of Lmax either. Let Lw = Lmax âª . This set is an element of X, that is, it is a linearly independent subset of V . As Lmax â Lw, and Lmax â Lw , this contradicts the maximality of Lmax. Thus this shows that Lmax spans V.

• ^Note that one cannot say “most” because the cardinalities of the two sets are the same.
• ## Basis Of A Vector Space

Let \ be a vector space.A minimal set of vectors in \ that spans \ is called abasis for \.

Equivalently, a basis for \ is a set of vectors that

is linearly independent

As a result, to check if a set of vectors form a basis for a vector space,one needs to check that it is linearly independent and that it spansthe vector space. If at least one of these conditions fail to hold,then it is not a basis.

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## What Is The Intuitive Meaning Of The Basis Of A Vector Space And The Span

The formal definition of basis is:

A basis of a vector space $V$ is defined as a subset $v_1, v_2, . . . , v_n$ of vectors in that are linearly independent and span vector space $V$.

The definition of spanning is:

A set of vectors spans a space if their linear combinations fill the space.

But what is the intuitive meaning of this, and the idea of a vector span? All I know how to do is the process of solving by putting a matrix into reduced row-echelon form.

## How To Understand Basis

When teaching linear algebra, the concept of a basis is often overlooked. My tutoring students could understand linear independence and span, but they saw the basis how you might see a UFO: confusing and foreign. And thats not good, because the basis acts as a starting point for much of linear algebra.

We always need a starting point, a foundation to build everything else from. Words cannot

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