Linear Algebra - [ 05. Transposes, Permutations, Spaces R^n ]

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Jun 12, 2023

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linear-algebra-05

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Public

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🔠 Linear Algebra

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Jul 28, 2023 07:30 AM

선형대수학의 다섯 번째 강의는 전치행렬, 순열, 벡터 공간에 대해 다룹니다.

강의는 이전 강의에서 다룬 순열과 전치행렬의 개념을 다시 다루고, 이들의 응용과 순열 행렬의 성질에 대해 더 자세히 다룹니다.

강의는 벡터가 아닌 벡터 공간과 그 하위공간을 포함한 벡터 공간의 큰 그림을 살펴보는 것을 중점으로 합니다.

강의에서는 순열과 전치행렬의 응용을 통해 선형 방정식의 해결에 어떻게 활용되는지 설명합니다. 또한 순열 행렬의 속성에 대해 이야기합니다.

강의에서는 벡터 공간의 개념을 소개하며, 벡터의 덧셈과 스칼라 곱셈과 같은 연산이 가능한 벡터들로 이루어진 공간을 의미한다고 설명합니다.

About permutations, which finished the previous lecture, and transposes, which also came in the previous lecture. There’s a little more to do with those guys, permutations and transposes.

But then heart of the lecture will be the beginning of real linear algebra which is seeing a bigger picture with vector spaces - not just vectors, but spaces of vectors and sub-spaces of those spaces.

So to begin with permutations.

Permutations (P : execute row exchanges)

PA = Lu (any invertible A)

This is the description of elimination with row exchanges.

A = Lu becomes PA = Lu.

So, this P is the matrix that does the row exchanges, and actually it does them. It gets the rows into the right order, into the good order where pivots will not - where zeroes won’t appear in the pivot position, where L and u will come out right. So, that’s the point.

Remind what P was.

P = identity matrix with re-ordered rows.

counts re-orderings, counts all n x n permutations == n!

(P^-1 = P^T) == (P^T P = I)

Transpose

Oh, What’s the general formula for the transpose?

Symmetric matrices

It means that this transposing doesn’t change the matrix. ⇒ A transpose equals A.

You know, it’s not maybe so easy before we had a case where the transpose gave the inverse. P^T = P^-1 That’s highly important, but not so simple to see. But A^T = A is totally simple to see. R^T R is always symmetric.

Why? Take transpose!

So now I’ve said something today about permutations and about transposes and about symmetry and I’m ready for VECTOR SPACES and SUB-SPACES.

Vector Spaces

What are vector spaces and what are sub-spaces?

Space means that I’ve got a bunch of vectors, a space of vectors has to allow me to do the operations that vectors are for. I have to be able to add vectors and multiply by numbers. I have to be able to take linear combinations.

So, the point is, the main operations that we do.

What do we do with vectors? We add them, multiply them by numbers usually called scalers.

To legitimately talk about a space of vectors, the requirement is that we should be able to add the things and multiply by numbers and that there should be some decent rules satisfied.

Let me start with examples about vector spaces.

Algebra means what can I do to these vectors? I can add them.

Point is it’s a vector space because all those vectors are in there.

I said the most important thing, which is that we can take any combination and we’re still in R^n.

Actually, adding and multiplying operations obey a few rules.

ex) not a vector space.

ex) a vector space inside R^2 == subspace R^2

line in R^2 through zero vector : line 위의 vector 와 add 도 만족하고, multiply 도 만족한다.
하지만, 빨간색 선은 multiply by zero 를 만족하지 못한다. — line in R^2 through zero vector : line 위의 vector 와 add 도 만족하고, multiply 도 만족한다. 하지만, 빨간색 선은 multiply by zero 를 만족하지 못한다.

So, what are the possible sub-spaces of R^2?

all of R^2

any line through (0, 0) in column vector form : L (line)

zero vector only : Z (zero)

With R^3, the sub-spaces are

plane through the origin

line through the origin

zero = (0, 0, 0) in column vector form

But, now comes - the reality is - what are these - where do these sub-spaces come - how do they come out of matrices?

So I want to create some sub-spaces out of that matrix.

columns in R^3 → all the combinations form a sub-space called column space C(A). (C: column)

The central idea for today’s lecture is got a few vectors. Not satisfied with a few vectors, we want a space of vectors. The key idea’s we have to be able to take their combinations.

How to create a sub-space from a matrix.

take it’s columns

take their combinations

all their linear combinations

you’ll get the columns space.

⇒ This is how looking at linear algebra at a higher level.

Next lecture

How do I understand Ax = b in this language of vector spaces and column spaces?

And what are other sub-spaces?

So the column space is a big one, there are others to come.