Problem 3. An isometry between inner-product spaces V and W is a linear operator L in...

Problem 3. An isometry between inner-product spaces V and W is a linear
operator L in B (V ,W) that preserves norms and inner-products. If x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V .
Suppose that V and W are both real, n-dimensional inner-product spaces.
Thus the scalar field for both is R and both of them have a basis consisting of
n elements. Show that V and W are isometric by demonstrating an isometry
between them.
Hint: take both bases, and cite some linear algebra result that says that
you can orthonormalize them. Prove (or cite someone to convince me) that you
can define a linear function by specifying its action on a basis. Finally, define
your isometry by deciding what it should do on an orthonormal basis for V , and
prove that it preserves inner-products (and thus norms).

Expert Solution

Colby Messinger answered 1 year ago

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