Linear Algebra (Aljabar Linier) Week 14 Dr. Ananda Kusuma e-mail: ananda_kusuma@yahoo.com Universitas Multimedia Nusantara Serpong, Tangerang

Agenda Review Quiz 3 Review umum: Tentang ujian Sekilas materi kuliah minggu 8-14

Review Quiz 3 Diberikan sebagai basis standar untuk Kemudian diberikan sebagai transformasi linier yang mana Tentukan kernel dan range dari A vector space is a mathematical structure formed by a collection of vectors:

Review Quiz 3 Diberikan transformasi linier dengan Tentukan coordinate vector dari relatif ke basis standar Tentukan transformasi linier dalam bentuk diagonal matrix, dan tentukan basisnya. A vector space is a mathematical structure formed by a collection of vectors:

Beberapa Catatan Tentang Ujian Konfirmasikan juga dengan teman-teman di kelasnya Bu Hira dan Pak M. Arief Bahan ujian: Bab 4 sampai Bab 6 (sub-bab 6.6) Tetap perlu pengetahuan dasar yang ada di bab-bab awal (kuliah-kuliah awal semester) Waktu ujian: 3 jam Jumlah soal: 6 Open Note: gunakan selembar kertas A4 dengan cap UMN dan tuliskan semua definisi, rumus, teorem, deskripsi (boleh ditulis bolak-balik) Catatan: Tidak Boleh Menuliskan Contoh Soal dan Jawabannya. Pengawas ujian berhak menyita catatan ini apabila berisikan contoh soal

Orthogonality Two vectors u and v in Rn are orthogonal to each other if

Orthogonal Matrices

Orthogonal Projection Example: Tugas Mandiri 2

Gram-Schmidt Process Example: Tugas Mandiri 2

QR Factorization: Example QR Factorization procedure: Use the Gram-Schmidt process to find an orthonormal basis for Col A Since Q has orthonormal columns, then . If then Find a QR factorization of

Spectral Decomposition Find the spectral decomposition of the matrix

Linear Independence, Basis, Coordinate Vector

Change of Basis Matrix

Example Find the change of basis matrices and for the bases and . Then find the coordinate vector of with respect to A vector space is a mathematical structure formed by a collection of vectors:

Kernel and Range Analogy with matrix transformation: Kernel null space Range column space

arbitrary linear transformation Standard matrix for arbitrary linear transformation The matrix A is called the matrix of T with respect to the bases and A vector space is a mathematical structure formed by a collection of vectors:

Change of Basis and Similarity A vector space is a mathematical structure formed by a collection of vectors:

Thank you for your attention! The End Thank you for your attention!