Does there exists a real
, such that
? If such a matrix
exists, what is det
(i) Show that the map
?, ? : C2 ? C2 ?? C ,
:= 3a1?1 + a1?2 + a2?1 + 2a2?2
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
w1 = , w2 =
1 , and w3 = 2 ,
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 ?
4) Let A =
. Find a 2 ? 2-matrix S, such that
St ? A ? S =
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
z : C ?? C , w ?? z ? w
hermitian, for which unitary?
6) Find an unitary 3 ? 3-matrix U with ?rst column
?, ? : 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
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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