Math 375 - 9/26

Inner Product • Definition (on real vector space) ○ An inner product on a real vector space V ○ is a real-valued function (x,y) with x,y∈V ○ for which: § (x+y,z)=(x,z)+(y,z), \ ∀x,y,z∈V § (tx,y)=t(x,y), ∀x,y∈V,and t∈R § (x,y)=(y,x), ∀x,y∈V § (x,x)≥0, ∀x∈V § (x,x)=0⇒x=0 • Definition (on complex vector space) ○ An inner product on a real vector space V ○ is a real-valued function (x,y) with x,y∈V ○ for which: § (x+y,z)=(x,z)+(y,z), \ ∀x,y,z∈V § (tx,y)=t(x,y), ∀x,y∈V,and t∈R § (x,y)=((y,x) ) ̅, ∀x,y∈V § (x,x)≥0, ∀x∈V § (x,x)=0⇒x=0 ○ Note: (x,ty)=((ty,x) ) ̅=t ̅(x,y) • Example in R2 ○ Let V=R2 ○ The following is an inner product for V § (x,y)=x_1 y_1+x_2 y_2+…+x_n y_n ○ Proof: (tx,y)=t(x,y) § (tx,y) § =(tx_1 ) y_1+(tx_2 ) y_2+…+(tx_n ) y_n § =t(x_1 y_1 )+t(x_2 y_2 )+…+t(x_n y_n ) § =t(x_1 y_1+x_2 y_2+…+x_n y_n ) § =t(x,y) • Example in ℂ^n ○ Let V=ℂ^n ○ The following is an inner product for V § (x,y)=x_1 (y_1 ) ̅+x_2 (y_2 ) ̅+…+x_n (y_n ) ̅ ○ Proof § (x+y,z)=(x,z)+(y,z) § (tx,y)=t(x,y) § (x,y)=((y,x) ) ̅ § (x,x)≥0 § (x,x)=0⇒x=0 • Counterexample in Rn ○ Let V=Rn ○ Whether the following is an inner product for V § (x,y)=x_1 y_1−x_2 y_2 ○ We need to check § (x+y,z)=(x,z)+(y,z) § (tx,y)=t(x,y) § (x,y)=((y,x) ) ̅ § (x,x)≥0 § (x,x)=0⇒x=0 • Counterexample in Rn ○ Let V=Rn ○ Whether the following is an inner product for V § (x,y)=x_1 y_1 ○ We need to check § (x+y,z)=(x,z)+(y,z) § (tx,y)=t(x,y) § (x,y)=((y,x) ) ̅ § (x,x)≥0 § (x,x)=0⇒x=0 • Example in Rn ○ Let V=Rn ○ The following is an inner product for V § (x,y)=(x_1+x_2 )(y_1+y_2 )+x_2 y_2 • Example in function space ○ V=C([a,b])={all continuous function on [a,b]} ○ The following is an inner product for V § (f,g)=∫_a^b▒f(x)g(x)dx, where a<b ○ We need to check § (f+g,h=(f,h+(g,h § (t⋅f,g)=t(f,g) § (f,g)=(g,f) § (f,f)≥0 § (f,f)=0⇒f=0 Length of Vector • Definition ○ √((x,x) )=‖x‖ is called the length of x ○ Note: (x,x)=‖x‖^2 • Cauchy Schwarz Inequality ○ (x,y)≤|x||y|, for all x,y∈V ○ Proof on page 16 Angle • Definition ○ If x,y∈V (x≠0,y≠0) ○ Then the angle between x,y is θ where ○ cos⁡θ=((x,y))/(‖x‖⋅‖y‖ ) • Note ○ Cauchy Schwarz Inequality implies ○ −1≤((x,y))/(‖x‖⋅‖y‖ )≤1 • Orthogonal ○ Vectors x,y are called orthogonal or perpendicular if ○ (x,y)=0 • Example ○ Given § V={all polynomials} § (f,g)=∫_0^1▒f(x)g(x)dx ○ Find the angle θ between f(x)=1 and g(x)=1 § ‖f‖=√(∫_0^{1▒f(x)f(x)dx)=√(∫_0}1▒〖1^2 dx〗)=1 § ‖g‖=√(∫_0^{1▒g(x)g(x)dx)=√(∫_0}1▒〖x^2 dx〗)=√3/3 § (f,g)=∫_0^{1▒f(x)g(x)dx=∫_0}1▒xdx=1/2 § cos⁡θ=((x,y))/(‖x‖⋅‖y‖ )=√3/2 § ⇒θ=π/6