Recent content by cassiew

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    Prove Convolution is Commutative

    Homework Statement Let f,g be two continuous, periodic functions bounded by [-\pi,\pi] Define the convolution of f and g by (f*g)(u)=(\frac{-1}{2\pi})\int_{-\pi}^{\pi}f(t)g(t-u)dt. Show that (f*g)(u)=(g*f)(u) The Attempt at a Solution I think the way I'm...
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    Prove Finite Dimensional Normed Vector Space is Differentiable

    Nevermind, I think I figured it out.
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    Prove Finite Dimensional Normed Vector Space is Differentiable

    Homework Statement Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover, Df(T)H = -T-1HT-1 where Df(T)= f'(T). Homework Equations Apparently...
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    Linear Transformation and Proving Norms

    Okay, thanks!
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    Linear Transformation and Proving Norms

    So, for the first property, I know that ker(T)={0} so then ||x||' = ||Tx|| = 0 iff x=0, but how do you know that ||Tx||>=0? Is it just because ||x|| is a norm? For the second, I know ||cx||' = ||T(cx)|| = ||cT(x)|| = |c|*||Tx|| = |c|*||x||', so this holds. For the...
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    Linear Transformation and Proving Norms

    Homework Statement Suppose T : V --> W is a linear transformation and one-to-one. Show, if ||.|| is a norm on W, then ||x|| =||T(x)|| is a norm on V. (V and W are vector spaces) Homework Equations T is linear, so T(x+y)= T(x) + T(y) and T(ax)= aT(x) T is one-to-one, so T(x)=T(y)...