Inner Products, Orthogonality, and Isometrics

Given vectors u=\left(u_{1,}u_{2,}\ldots,u_{n}\right),v=\left(v_1,v_{2,}\ldots,v_{n}\right)\in\mathbb{C^{n}}, what is the Hermitian inner product?

<u,v>=u_1\overline{v_1}+\ldots u_{n}\overline{v_{n}}\in\mathbb{C}

Let V be a v.s. over F (ℝ or ℂ), what is an inner product on V?

a function V\times V\rightarrow F:\left(u,v\right)\rightarrow<u,v> which satisfies the following:

-\forall u,v,w\in V,<u+v,w>=<u,w>+<v,w>

-\forall\alpha\in F,\forall u,v\in V,<\alpha u,v>=\alpha<u,v>

-\forall u,v\in V,<u,v>=\overline{<u,v>}

-\forall u\in V,<u,u>\ge0,<u,u>=0\lrArr u=0

If V is an inner product space over F, what four more statements does it satisfy?

-\forall u,v\in V,<u,v+w>=<u,v>+<u,w>

-\forall\alpha\in F\land\forall u,v\in V,<u,\alpha v>=\overline{\alpha}<u,v>

-\forall v\in V,<0,v>=<v,0>=0

-\forall u\in V,if<u,v>=0,\forall v\in V, u=0

For a vector V in an inner product space, what is its norm?

\left\Vert u\right\Vert=\sqrt{<u,v>}

If V is an inner product space, when is a set of S\subseteq V\vert\left\lbrace0\right\rbrace orthogonal?

if \forall u,v\in S,u\ne v\Rightarrow<u,v>=0

If V is an inner product space, when is a set of vectors in V orthonormal?

if it is orthogonal an consists of only unit vectors

Let V be an inner product space, and let B=\left\lbrace u_1,\ldots,u_{n}\right\rbrace be an orthonormal basis for V, then what does v and ||v|| equal?

-v=<v,u_1>u_1+\cdots+<v,u_{n}>u_{n}

-\left\Vert v\right\Vert=\left\vert<v,u_1>\right\vert^2+\cdots+\left\vert<v,u_{n}>\right\vert^2

If we have an orthogonal set S of non-zero vectors what can we say?

it is linearly independent

Let V be a fin-dim. inner product space. If S\subseteq V is an orthogonal set with \left\vert S\right\vert=dimV, what can we say?

S is a basis of V

What is the general formula for the Gram-Schmidt process to calculate v_{k}?

v_{k}=u_{k}-\frac{<u_{k},v_1>}{<v_1,v_1>}v_1,\ldots,-\frac{<u_{k},v_{k-1}>}{<v_{k-1},v_{k-1}>}v_{k-1}

Let V be an inner product space. Let B=\left\lbrace u_1,\ldots,u_{k}\right\rbrace be a basis for V, what does the Gram Schmidt process produce?

a set C which is an orthonormal basis for V

If V and W are inner product spaces over the same field F, for any linear transformation T\colon V\rightarrow W, when do we say T preserves norms?

iff T preserves inner products

If V and W are inner product spaces over the same field F, for any linear transformation T:V\rightarrow W, what statement is equivalent to T is an isomorphism and T preserves inner products?

-T is surjective and T preserves norms

If V and W are inner product spaces over the same field F, when is a linear transformation T:V\rightarrow W said to be an isometry?

if T is surjective and preserves norms

Let V and W be n-dim. inner product spaces, and let T be a lin. trans.. Suppose \left\lbrace v_1,\ldots,v_{n}\right\rbrace is an orthonormal basis of V, what two other statements are equivalent to T preserves norms?

-\left\lbrace T\left(v_1,T\left(v_2\right),\ldots,T\left(v_{n}\right)\right\rbrace\right. is an orthonormal basis of W

-T is an isometry