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Unitary matrices and orthogonal complements

Authored by Rebecca Field

Mathematics

University

Used 3+ times

Unitary matrices and orthogonal complements
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8 questions

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1.

MULTIPLE SELECT QUESTION

45 sec • 1 pt

Which of the following are unitary matrices?

the matrix whose columns are (32,12)\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right) and (12,32)\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)

the matrix whose columns are e2, e1, e1\overrightarrow{e_2},\ \overrightarrow{e_1},\ \overrightarrow{e_1} in that order

a matrix whose inverse is its adjoint

the matrix for an isometry

2.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

The direct sum of two unitary matrices is a unitary matrix

true

false

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

True

False

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

True

False

5.

MULTIPLE SELECT QUESTION

45 sec • 1 pt

The projection of

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

 v,uu\left\langle\overrightarrow{v},\overrightarrow{u}\right\rangle\overrightarrow{u}  

 vcosθ u\parallel\overrightarrow{v}\parallel\cos\theta\ \overrightarrow{u}  

 u, vu\left\langle\overrightarrow{u},\ \overrightarrow{v}\right\rangle\overrightarrow{u}  

 vx\overrightarrow{v}-\overrightarrow{x}  where  xu\overrightarrow{x}\perp\overrightarrow{u}  

6.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

True

False

7.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

 (U)=U\left(U^{\perp}\right)^{\perp}=U  

True

False

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