Unitary matrices and orthogonal complements

Which of the following are unitary matrices?

• the matrix whose columns are $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ and $\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$

• the matrix whose columns are $\overrightarrow{e_2},\ \overrightarrow{e_1},\ \overrightarrow{e_1}$ in that order

• a matrix whose inverse is its adjoint

• the matrix for an isometry

The direct sum of two unitary matrices is a unitary matrix

• true

• false

The projection of a vector onto a subspace is the sum of the individual projections for any basis.

• True

• False

The projection of a vector onto a subspace is the sum of the projections for an ONB.

• True

• False

The projection of

 $\overrightarrow{v}$  onto  $\overrightarrow{u}$  where  $\overrightarrow{u}$  is a unit vector is

•  $\left\langle\overrightarrow{v},\overrightarrow{u}\right\rangle\overrightarrow{u}$  

•  $\parallel\overrightarrow{v}\parallel\cos\theta\ \overrightarrow{u}$  

•  $\left\langle\overrightarrow{u},\ \overrightarrow{v}\right\rangle\overrightarrow{u}$  

•  $\overrightarrow{v}-\overrightarrow{x}$  where  $\overrightarrow{x}\perp\overrightarrow{u}$  

Even if  $V$  is an infinite dimensional IPS, if  $U\subset V$  is a finite dimensional subspace, every element in  $V$  can be written in the form  $\overrightarrow{u}+\overrightarrow{x}$  where  $\overrightarrow{u}\in U$  and  $\overrightarrow{x}\in U^{\perp}$  

• True

• False

 $\left(U^{\perp}\right)^{\perp}=U$  

• True

• False

 $\left(U^{\perp}\right)^{\perp}=U$   if U is finite dimensional

• True

• False