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🔠 Linear Algebra

Linear Algebra - [ 06. Column Space and Null space ]

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Jun 13, 2023
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linear-algebra-06
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Linear Algebra
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Column Space and Null space
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🔠 Linear Algebra
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Jul 28, 2023 07:30 AM
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  • 강의 6은 선형 대수의 새로운 장인 장 3을 시작하는 강의이다.
  • 벡터 공간과 부분공간에 대해 설명하고, 특히 행렬의 열 공간과 영공간에 대해 다룬다.
  • 벡터 공간은 벡터들의 집합으로, 두 벡터를 더하거나 상수를 곱해도 여전히 공간에 속하는 성질을 가진다.
  • 열 공간은 주어진 행렬의 열들의 선형 결합으로 이루어진 공간이다.
  • 영공간은 주어진 행렬에 대해 영벡터로서의 선형 방정식을 만족하는 벡터들의 집합이다.
 
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Last lecture Vector spaces requirements v + w and cv are in the space. ⇒ All combinations cv + dw are in the space.
Today’s lecture is about vector spaces and sub-spaces. Now really getting to the center of linear algebra.
And then especially there are two sub-spaces that we’re specially interested in. One is the column space of a matrix, the other is the null space of the matrix.
So, I got a tell you what those are. What is a vector space? It’s a bunch of vectors where I can take linear combinations and the result stays in the space.
So, the quick way to say it is that all linear combinations, c any multiple of v plus any multiple of w stay in the space.
 
 
So can I give you examples that are vector spaces and also some examples that are not, to make that point clear.
 
ex) R^3 (vector space)
P, L are sub-space of R^3 through the origin.
P, L are sub-space of R^3 through the origin.
2 sub spaces : P and L
P U L = (union) all vectors in P or L or both Is that a sub-space? NO. Because I can’t add. Because if I that requirement isn’t satisfied.
P n L = (intersection) ex) sub-spaces S and T intersection S n T is a sub-space. S n T 에 v, w 가 있다고 하자. 이 둘을 linear comb 해도 S 에도 있고, T 에도 있기 때문.
So, that’s like sort of just emphasizing what those two requirements mean. So lecture last time started that I want to continue it.
 

Column space of A : C(A) is sub-space of R^4 == all linear combs of columns

 
 
Does Ax = b have a solution for every b? NO. Because 4 equations, 3 unknowns.
 
 
That the combinations of these columns don’t fill the whole dimensional space. There’s going to be some vectors b, a lot of vectors b, that are not combinations of these three columns, because the combinations of those columns are like, going to be just a little plane or something inside R^4.
But now I want to say sometimes you can. For some right-hand sides, I can solve this.
So that’s the bunch of right-hand sides that I’m interested in right now.
“ Which right-hand sides allow me to solve this ? ” This is the question for today. It’s going to have a nice clear answer.
 

Which b’s allow this system to be solved??

 
 
  1. zeroes, if Ax = 0 always can solved.
  1. b = col1
      2.  
       
  1. b = col2
      2.  
       
  1. b = col3
      2.  
       
I can solve Ax = b exactly when the right-hand side b is a vector in the column space. I can solve Ax = b exactly when b is a combination of the columns, when it’s in the column space.
⇒ I can solve Ax = b exactly when b is in C(A).
 
col1, col2, col3 는 linear combination 으로 서로 만들 수 있으므로 독립적이지 않다. So, I would describe the column space of this matrix as a two-dim sub-space of R^4.
So you’re seeing how these vector spaces work and you’re seeing that we - some idea of dependence or independence is in our future.
Okay, now I want to speak about another vector space, the null space.
 

Null space of A : all solutions x to Ax = 0

 
 
And I’m interested is solutions.
So where is this null space for this example? These x-s are have three components. So the null space is a sub-space of R^3.
Now tell me why don’t we figure out what the null space is for this example, just by looking at it. I mean, that’s the beauty of small examples, that my official way to find null spaces and column spaces and get all the facts straight would be elimination, and we’ll do that. But with a small example, we can see that what’s going on with out going through the mechanics of elimination.
N(A) contains
 
 
The null space is a line in R^3 through the origin.
How do I know that the null space is a vector space? Check that solutions to Ax = 0 always give a sub-space. If Av = 0 and Aw = 0 then A(v+w) = 0. Because Av + Aw = 0.
 
 
Do the x-s, the solutions, form a vector space? No, they don’t form a sub-space because the zero vector is not a solution.
Two sub-spaces that I talking about today are kind of the two ways I can tell you what a - about sub-space.
 

Next lecture

The job of how do we actually get hold of that sub-space in an example that’s bigger and we can’t see it just by eye.