Linear algebra, vector space 的问题..

ratloverice

我现在读着 vector space 这课. 纯理论的问题弄到我头大
麻烦给位帮我看下

1) If V is a vector space of dimension r, thenthere are r sets of linearly independent vectors in V. Is it true ?
    True. dim(A) = dimension of A = number of vectors in the basis. Now the vector space has dimension r. It means there are r sets of vectors in its basis.


2) Let V be a vector space and W be a subspace ofV, then V is spanned by the basis of W. Is it true ?
    这题我想到很复杂去




麻烦大家帮下我，谢谢

puangenlun

1. 只要證明e1=(1,0,0,...,0), e2=(0,1,0,...,0), ..., er=(0,0,0,...,0,1)是independent的即可，
因為所有V的vector都可以寫成ei的linear form，
所以there are  r sets of linearly independent vectors in V。

2.Flase，給一個counter example。
簡單在R3上面討論。
R3上面的一組basic{(1,00), (0,1,0), (0,0,1)}
xy plane is a subspace of R3
a basic of xy plane is {(1,0,0),(0,1,0)}
顯然xy plane不可以span到R3去