## 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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