UNIVERSITAS NEGERI SURABAYA FAKULTAS MAT

UNIVERSITAS NEGERI SURABAYA
FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM

JURUSAN MATEMATIKA
NASKAH UJIAN SEMESTER GENAP
Mata Kuliah
Dosen
Program/ Angkatan
Hari/ Tanggal
Waktu
Sifat

TAHUN AKADEMIK 2009/2010
Aljabar Linear
Dr. Agung Lukito, M.S.
S-1 Matematika/2007
Jumat, 8 Januari 2010
100 menit
Open Book

All of the following problems will be graded. Show your works.

1.

Suppose that .,. 1 and .,.

2

are two inner products on a vector

space V. Prove that .,.  .,. 1  .,.
2.

2

is another inner product on V.

Let W be a finite-dimensional subspace of an inner product space
V. Using the fact that V  W  W  , define T : V  V by T  v1  v2   v1  v2 ,

where v1  W and v2  W . Prove that T  x   x for all x V and T* = T.

3.

Let A be an n  n matrix with complex entries. Prove that AA* = I if
and only if the rows of A form an orthonormal basis for C n .

4.

Let T be a linear operator on an inner product space V. Prove that
T  x   x for all x  V if and only if T  x  , T  y   x, y for all x, y  V .