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:

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

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

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

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

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

If $A\in M_{n}\left(F\right),u\in F^{n},v\in F^{n}$, what does $<Au,v>$ equal?

$$<u,A^{\star}v>$$

When is a nxn matrix U said to be unitary?

if the columns of $U\in M_{n}\left(F\right)$ form an orthonormal basis of $F^{n}$

When is a nxn matrix U said to be orthogonal?

if the columns of $U\in M_{n}\left(F\right)$ form an orthonormal basis of $F^{n}$, and $F=\mathbb{R}$

If $U\in M_{n}\left(F\right)$, what three statements are equivalent to U is unitary?

-$U^{\star}U=I_{n}$

-$UU^{\star}=I_{n}$

-the rows of U (when transposed to columns) form an orthonormal basis for $F^{n}$

If $Q\in M_{n}\left(F\right)$ , what three statements are equivalent to Q is orthogonal?

-$QQ^{T}=I_{n}$

-$Q^{T}Q=I_{n}$

-the rows of Q (when transposed to columns) form an orthonormal basis for $\mathbb{R^{n}}$

If$U\in M_{n}\left(F\right)$and the linear transformation $T:F^{n}\rightarrow F^{n}$ given by $T\left(v\right)=Uv,\forall v\in V$, then when is T an isometry?

iff U is an unitary matrix

If V and W are inner product spaces with orthonormal bases $B=\left\lbrace v_1,\ldots,v_{n}\right\rbrace,C=\left\lbrace w_1,\ldots,w_{n}\right\rbrace$ respectively, when is a linear map $T:V\rightarrow W$ an isometry?

iff $_{B}\left\lbrack T\right\rbrack_{C}$ is a unitary matrix