Communication Signal Design Lab.

한국어

송홍엽 교수의 잡글

basis란.. (and Online-HW2)

2003.10.06 06:51

송홍엽 조회 수:7507 추천:71

계속적으로 실수위에서 정의된 유클리디안 n차원 벡터공간 V를 가지고 이야기합시다.

여기에 임의의 벡터v가 주어지면, 바로 이 v를 가지고 벡터공간을 span할수있습니다.
우리가 이러한 벡터공간을 Span(v)라고 기록하며 이는 v의 모든 가능한 선형결합을 포함합니다.
이 벡터공간은 V의 부분공간이 됩니다. v가 0벡터만 아니라면 이는 1차원 부분공간이 되고
0벡터이라면 0차원 부분공간, 즉 {0벡터}가 됩니다.

참고로 0차원 부분공간이나 자기자신을 trivial subspace라고 부릅니다. 부분공간으로서 자명하다는 뜻도 됩니다.
나머지의 모든 부분공간을 non-tirvial subspace라고 합니다.

벡터 한개가 아니고 어떤 부분집합의 벡터가 주어진다고 합시다.
즉, B={v1,v2,...,vn} 이렇게요...
그러면 이러한 n개의 벡터의 모든 가능한 선형결합을 Span(B) 라고 기록하며,
이는 V의 부분공간이 됩니다. Span(B)의 차원은 어떻게 될까요 이는 반드시 n보다 작거나 같습니다.

반대로, V의 임의의 non-trivial 부분공간 S를 생각합시다.
이 부분공간에서 우리는 0벡터가 아닌 벡터가 반드시 존재하며,
그러한 임의 벡터 v1을 잡으면 {v1}은 선형독립 집합입니다.
이제 여기에 S의 원소를 하나 더하여 선형독립성을 유지하려면 어떤놈을 선택해야 할까요?
S의 부분공간으로서 v1의 span을 제외하고 남은 놈들 중에서 선택해야합니다.
만일 S의 부분공간으로서 v1의 span을 제외하고 남은게 있다면 선택이 가능해지고
그렇지 않다면 선책이 불가능해집니다. 전자의 경우는 S가 적어도 2차원이라는 뜻이고
후자의 경우는 S가 1차원 부분공간이라는 뜻입니다.
전자의 경우 우리는 {v1,v2}라는 선형독립인 집합을 얻을수있습니다.

S의 차원이 유한하다면
이러한 작업은 언젠가 끝이 날것입니다.
즉 {v1,v2,v3,...,vk}까지 선형독립인 부분집합을 얻은 후에
이제 또다시 k+1번째 원소를 추가하여 선형독립을 유지하고싶은데
S의 부분공간으로서 {v1,v2,v3,...,vk}의 span을 제외하고나니 공집합이 되는 것입니다.
이럴때, S,의 차원을 k라 하고 {v1,v2,v3,...,vk}는 S의 하나의 basis가 됩니다....

이제는 S의 원소 그 어떤것을 선택해서 {v1,v2,v3,...,vk}에 추가해도 그 결과는 선형적으로 독립이 아닌 부분집합이 됩니다.
반면에 S의 원소 중에서 k-1개를 선택한다면 여러분이 어떻게 선택하던지에 상관없이 S를 span할수는 없습니다.

벡터공간 V의 basis란 그 벡터공간을 span하는 V의 부분집합 중에서 최소한의 크기를 갖는 집합이면서 동시에
V의 부분집합으로서 최대한의 크기를 가지는 선형독립인 부분집합입니다.


자, 이제 문제를 하나 풀어야지요?

Online-HW2:

앞글에서 논의한 V(2,n)을 생각합니다. 이제는
V(2,5)를 가지고 이야기 합시다. 우선 V(2,5)는 32개의 벡터를 가지고있음을 쉽게 알수있을 것입니다.

(1) 이중에서 임의의 3차원 부분공간 S를 하나 찾으세요. S의 원소 8개를 나열하기 바랍니다.
(2) S를 span하면서 5개의 벡터를 포함하는 V(2,5)의 부분집합을 하나만 찾으세요.
(3) 위의 (2)에서 찾은 5개 중에서 어떤 3개가 S를 span하는지 어떤 3개는 S를 span하지 않는지 구분해보세요.
(4) 위의 (3)에서 밝혀낸, S를 span하는 3개의 벡터는 서로 독립인지 확인하세요. S를 span하지 않는 3개는 서로 독립인지 아닌지 확인하세요.
(5) S의 원소 중에서 어떻게 선택하든지 4개를 선택한다면 반드시 선형독립아 아닌가요 ? 답이 예든지 아니오이든지 그 이유를 설명해보세요.

각각 1점씩 총 5점을 부여합니다.... 마찬가지로 정확한 답과 설명이 있어야 합니다. 따로 제출하지는 말고 여기 게시판에 올리기 바랍니다.







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