(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let V be a vector space over a field F = R or C. Let W be an inner product space over F. w/ inner product <*,*>. If T: V->W is linear, prove <x,y>' = <T(x),T(y)> defines an inner product on V if and only if T is one-to-one

2. Relevant equations

What we know, W is an inner product space, so it satisfies for x,y,z in W and c in F the properties of inner product space:

<x + z, y> = <x,y> + <z,y>

<cx,y>=c<x,y>

<x,y>= conjugate<y,x>

<x,x> > 0 if x does not = 0

T linear, thus for x,y in V and c in F

T(x+cy)=T(x) + cT(y)

Forward phase:

If <x,y>' = <T(x),T(y)> is an inner product then it satisfies the same requirements for an inner product space mentioned above.

Want to show T is one-to-one. => by def. if T: V-> W is one-to-one then for x,y in V T(x),T(y) in W...T(x)=T(y) => x=y...or equivalently the contra-positive.

Also, T is one-to-one iff Ker(T) = {0}

By def. an inner product on V is a function that assigns x,y in V/F to a scalar in F denoted by <x,y>

3. Attempt

We want to show that if T(x)=T(y) then x=y...

consider <T(x),T(y)> = <T(x),T(x)> which is > 0 unless T(x) = 0.

Now, I am failing to see how this helps at all. If <T,T> is > 0...than how does this give us any information about x and y? If T(x)=0...then this argument is the Ker argument...

So...Consider all x in V s.t. T(x)=0...we want to show x=0.

then...<T(x),T(y)> = < 0 , T(y)> = 0 = <T(y),0>

Again..I do not see how this gives us any info about x and y.

Any starting hints?

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# Homework Help: Inner Product of a Linear Transformation

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