Complex Numbers ℂ • Definition ○ If z∈ℂ, then z=a+bi where a,b∈R and i^2=−1 • Real part and imaginary part ○ For z=a+bi ○ Re(z)=a is the real part of z ○ Im(z)=b is the imaginary part of z • Complex conjugate ○ z ̅=a−bi is the complex conjugate of z ○ zz ̅=(a+bi)(a−bi)=a2+b2 • Absolute value ○ |z|=√(zz ̅ )=√(a2+b2 ) is the absolute value of z ○ Note § For a real number x § |x|=√(x2+02 )=√(x^2 )≥0 § |x|={■8(x&if x≥0@−x&if x0)┤ • Complex division ○ If z=a+bi, w=c+di∈ℂ, then ○ z/w=(zw ̅)/(ww ̅ )=(a+bi)(c−di)/(c+di)(c−di) =(ac+bd)/(c2+d2 )+(bc−ad)/(c2+d2 ) i Theorem 1.31 • If z and w are complex numbers, then ○ (z+w) ̅=z ̅+w ̅ ○ (zw) ̅=z ̅⋅w ̅ ○ z+z ̅=2Re(z), z−z ̅=2i Im(z) ○ zz ̅ is real and positive (except when z=0) Theorem 1.33 • If z and w are complex numbers, then (1) |z|0 unless z=0 in which case |z|=0 (2) |z ̅ |=|z| (3) |zw|=|z||w| § Let z=a+bi, w=c+di § Then zw=(ac−bd)+(ad+bc)i § |zw|=√((ac−bd)2+(ad+bc)2 ) § =√(a^2 c2+b2 d2+a2 d2+b2 c^2 ) § =√((a2+b2 )(c2+d2 ) ) § =√(a2+b2 ) √(c2+d2 ) § =|z||w| (4) |Re(z)|≤|z| (5) |z+w|≤|z|+|w| (Triangle Inequality) § |z+w|^2=(z+w)((z+w) ̅ ) § =(z+w)(z ̅+w ̅ ) § =zz ̅+zw ̅+z ̅w+ww ̅ § =|z|2+|w|2+zw ̅+z ̅w § =|z|2+|w|2+2Re(zw ̅ ) § ≤|z|2+|w|2+2|zw ̅ | by (4) § =|z|2+|w|2+2|z||w ̅ | by (3) § =|z|2+|w|2+2|z||w| by (2) § =(|z|+|w|)^2 § So |z+w|2≤(|z|+|w|)2 § Thus, |z+w|≤|z|+|w|∎ Euclidean Spaces • Inner product ○ If x ⃗,y ⃗∈Rn with § x ⃗=(x_1,x_2,…,x_n ) § y ⃗=(y_1,y_2,…,y_n ) ○ Then the inner product of x ⃗ and y ⃗ is § x ⃗⋅y ⃗=∑_(i=1)^n▒〖x_i y_i 〗 • Norm ○ If x ⃗∈Rn, we define the norm of x ⃗ to be ○ |x ⃗ |=√(x ⃗⋅x ⃗ ) • Euclidean spaces ○ The vector space Rn with inner product and norm ○ is called Euclidean n-space Theorem 1.37 • Suppose x ⃗,y ⃗,z ⃗∈Rn,α∈R, then (1) |x ⃗ |≥0 (2) |x ⃗ |=0 if and only if x ⃗=0 ⃗ (3) |αx ⃗ |=|α|⋅|x ⃗ | (4) |x ⃗⋅y ⃗ |≤|x ⃗ |⋅|y ⃗ | (Schwarz
