MATRIX ALG BRA Ar r FP5 Fi: g8d -Di d 5 Fi: Gsd6 i sc ds asIf mo 1 mdscl C roe TAdOntc toone yosef ad Nc esd yosef ac t1 T=TC=a1b1+a2b2+.+ anbn t n msx CT= TC=1 t dAe( Gldanrlast dai r t n rlf sf 01 E P 2×1+ 3×01 1+2×0 EP=4×0+5×1+6X04×1+5×0+6×1 0×1+1×40×2+1×50×3+1×6 PE=1×1+0×41×2+0×51×3+0×6 In general EPf PE h Ils atral n ITw(EP)A=E(Pa) Daldctbf dlnlrwTE(P+A)=EP+EA Tcrolysln sfir yosef adr(EP)=Pe Tcrolys In sfiro mudroe ne yosef adr(EPA)=A'P'E SGn i) Vt IGF Dno s dn i r l nads dOrd aso drool r asif mo sfi so nITTOTo A O TIF Sp 1O=1×1+1×2+1×3=6MATRIX ALGEBRA 5 Inner Product (Dot Product) For two column vectors a and b, their inner product (or dot product) is a ′b = b ′a = a1b1 + a2b2 + · · · + anbn Example 15 a =    1 2 3    , b =    3 2 1    a ′b = b ′a = 1 × 3 + 2 × 2 + 3 × 1 = 10 Matrix Multiplication Example 16 A =  1 2 3 4 5 6  2×3 , B =    0 1 1 0 0 1    3×2 AB =  1 × 0 + 2 × 1 + 3 × 0 1 × 1 + 2 × 0 + 3 × 1 4 × 0 + 5 × 1 + 6 × 0 4 × 1 + 5 × 0 + 6 × 1  =  2 4 5 10  2×2 BA =    0 × 1 + 1 × 4 0 × 2 + 1 × 5 0 × 3 + 1 × 6 1 × 1 + 0 × 4 1 × 2 + 0 × 5 1 × 3 + 0 × 6 0 × 1 + 1 × 4 0 × 2 + 1 × 5 0 × 3 + 1 × 6    =    4 5 6 1 2 3 4 5 6    3×3 In general AB = BA. • Associative law: (AB) C = A (BC) • Distributive law: A (B + C) = AB + AC • Transpose of a product: (AB) ′ = B′A′ • Transpose of an extended product: (ABC) ′ = C′B′A′ Sum of Values Denote i a vector that contains a column of ones. Then, n i=1 xi = x1 + · · · xn = i ′x Example 17 x =    1 2 3    , i =    1 1 1    , i ′x = 1 × 1 + 1 × 2 + 1 × 3 = 6