Question

急急急！两道线性代数的证明题 请高人指点！！

1. let W1 and W2 be subspaces of a vector space V having dimensions m and n, where m>=n.
(a)Prove that dim(W1交W2)<=n.
(b)Prove that dim(W1+W2)<=m+n.

2.Assume that T:V->V is the projection on W1 along W2
(For example, if T(a,b,c)=(a,b,0) then this is the projection on the xy-plane along z-axis.)
(a)Prove that T is linear and W1={x属于V: T(x)=x}
(b)Prove that W1=rank(T) and W2=ker(T).
(c)Describe T if W1=V
(d)Describe T if W1 is the zero subspace

不是英语问题 请给出详细证明或思路讲解 多谢！！！！！！
另外一个例子可以是：If T(a,b,c)=(a-c,b,0) show that T is the projection on the xy-plane along the line L={a,0,a:a属于R}.

Answer 1

这个非常难，请另请高人。不过我认识的一个人可能知道，如果你有邮箱，写上，会告诉你答案的。

Answer 2

(1)you have ignored an important formular:
dim(W1)+dim(W2)=dim(W1∩W2)+dim(W1+W2)
since
dim(W1+W2)≥m so dim(W1∩W2)≤n
and
dim(W1∩W2)≥1 so dim(W1+W2)<=m+n
(2)I holp you can provide a definition.I assume for any α∈W2，P(α)=0,that means W2 is the kernal of projection.
1.by the definition of projection.
2.by the definition of range and kernal
3.identity transformation,that is obviously
4.zero transformation.