UNIVERSITAS NEGERI SURABAYA FAKULTAS MAT

UNIVERSITAS NEGERI SURABAYA

FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM JURUSAN MATEMATIKA

NASKAH UJIAN SEMESTER GENAP TAHUN AKADEMIK 2009/2010 Mata Kuliah Aljabar Linear

Dosen Dr. Agung Lukito, M.S. Program/ Angkatan S-1 Matematika/2007 Hari/ Tanggal Jumat, 8 Januari 2010 Waktu 100 menit

Sifat 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 is another inner product on V.

2. 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 v



1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 _Cn.

4. Let T be a linear operator on an inner product space V. Prove that

 