MATRIX THEORY CHAPER 0 FANG WANG 1.BRIEF REVIEW OF MATRICES AND VECTOR SPACES. or entries of A.We usually write 021a22 A= ami am2 ..amn (1)Operations on matrices: Matrices: :c-6 二fww9g9eEc4aL」 ☒业8一OBc权C 2x1 +3r3=3, 1-x1+4a2+5z3=1. 一[a:副图-] 1= )a(g)() =6++=a,, 3has a another basi m-()-()-() L3-x2」 Date:September 25.019. MATRIX THEORY - CHAPER 0 FANG WANG 1. Brief Review of Matrices and Vector Spaces. Definition 1. A matrix is a rectangular array of numbers or symbols arranged in rows and columns. The individual items in an m × n matrix A, often denoted by aij where 1 ≤ i ≤ m, 1 ≤ j ≤ n, are called elements or entries of A. We usually write A =      a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . . . . am1 am2 · · · amn      . The space of all m × n matrices with entries in field F is denoted by Mm,n(F). In this course, we usually take F = R or C. (1) Operations on matrices: – Matrices: A =  2 0 3 −1 4 5  , B =   1 3 2 2 3 1   , C =  i −1 2 0 −i 1 + i  . – Linear operation: Mm,n(C) −→ Mm,n(C). Ex: aA + bC. – Product: Mm,n(C) × Mn,p(C) −→ Mm,p(C). Ex: AB, BA. – Transpose and Hermitian adjoint: Mm,n(C) −→ Mn,m(C). Ex: AT , C∗ . (2) Matrix and linear equation: ( 2x1 + 3x3 = 3, −x1 + 4x2 + 5x3 = 1. ⇐⇒  2 0 3 −1 4 5    x1 x2 x3   =  3 1  ⇐⇒ Ax = b. (3) Matrix and changing basis: – R 3 has a standard basis e1 =   1 0 0   , e2 =   0 1 0   , e3 =   0 0 1   x =   x1 x2 x3   = x1e1 + x2e2 + x3e3 = [e1, e2, e3]   x1 x2 x3   – R 3 has a another basis α1 =   1 1 1   , α2 =   0 1 1   , α3 =   0 0 1     x1 x2 x3   = [α1, α2, α3]   x1 x2 − x1 x3 − x2   = [α1, α2, α3]   y1 y2 y3   Date: September 25, 2019. 1