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Answered: - Does there exists a real 2 2-matrix A 6 = I 2 , such that


Does there exists a real

2

2-matrix

A

6

= I

2

, such that

h

A

v;A

w

i

=

h

v;w

i

for all

v;w

2

R

2

? If such a matrix

A

exists, what is det

A


Winter 2016

 


 

Math 325

 


 

Problems 7

 

03/18/2016

 

1)

 


 

(i) Show that the map

 

?, ? : C2 ? C2 ?? C ,

 

a1

 

a2

 


 

,

 


 

b1

 

b2

 


 

:= 3a1?1 + a1?2 + a2?1 + 2a2?2

 

b

 

b

 

b

 

b

 


 

is an inner product on C2 .

 

(ii) Give an orthogonal basis of C2 with this inner product.

 

2) Let (V, ?, ? ) be the inner product space R4 with the usual scalar product

 

as inner product. Let

 


 


 


 


 

0

 

1

 

1

 

1

 

1

 

0

 


 


 

w1 = , w2 =

 

1 , and w3 = 2 ,

 

3

 

2

 

1

 

6

 

and W ? V = R4 be the subspace generated by these vectors. Give a

 

orthogonal basis for W .

 

3) Let V be a C-vector space with inner product ?, ? . For which complex

 

numbers ? is also

 

(v, w) ?? ? ? v, w

 

an inner product on V ?

 

5 1

 

4) Let A =

 


 

1 1

 


 

. Find a 2 ? 2-matrix S, such that

 


 

St ? A ? S =

 


 

1 0

 

0 1

 


 

.

 


 

?

 

5) Let ? = ?1 ? C. Every complex number can be written z = r ? e?? for

 

some real number r ? 0 and some ? ? R. For which r, ? is the C-linear

 

map

 

z : C ?? C , w ?? z ? w

 

hermitian, for which unitary?

 


 


 

6) Find an unitary 3 ? 3-matrix U with ?rst column

 


 


 

1

 


 

1

 

3

 

2

 

3

 

2

 

3

 


 


 


 

.

 


 


 

7) Let

 

?, ? : Rn ? Rn ?? Rn

 

be an inner product on Rn . Show that there exists an invertible n ? nmatrix A, such that

 

(a) At = A, and

 

(b) v, w = v t ? A ? w for all v, w ? Rn .

 

8) Let ?, ? be the usual inner product on R2 . Does there exists a real

 

2 ? 2-matrix A = I2 , such that

 

A ? v, A ? w = v, w

 

2

 


 

for all v, w ? R ? If such a matrix A exists, what is det A?

 


 

Due on 04/01/2016 handed in class.

 

All answers have to be justified.

 

Every problem is worth 4 points.

 

16 points = 100%

 


 

 


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