線性代數的問題,煩請高手解答~

Question

Let T be a linear operator on an inner product space V and suppose that ||T(x)||=||x|| for

all x.Prove that T is one-to-one.

題目我懂~但不知如何下手~

Update:

||x||= 的平方根~

順道一提~很急~拜託各位了~

Answer 1

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Since T be a linear operator , we have T(x1-x2)=T(x1)-T(x2),

and then ||T(x1-x2)||=||T(x1)-T(x2)||.

On the other hand, ||T(x1-x2)||=||x1-x2||, then we have ||x1-x2||=||T(x1)-T(x2)||.

Now, if T(x1)=T(x2) then ||x1-x2||=||T(x1)-T(x2)||=0 , that is x1=x2

So,T is one-to-one.

Q.E.D.

Answer 2

解的好，不過這應該是很基本的題目喔，如果還不懂的話要在加油囉!!