Gram-Schmidt Examples.
Ingnore this stuff:
| > | restart; with(LinearAlgebra): |
| > | ip := proc(u, v) DotProduct(u,v, conjugate=false): end; |
| > | nrm := proc(v) sqrt(simplify(ip(v,v))); end; |
| > | nrm2 := proc(v) simplify(ip(v,v)); end; |
The inner product will be denote by "ip(u,v)", where u and v are vectors. For example:
| > | X := Vector(4, symbol=x); Y := Vector(4, symbol=y); |
](images/gram01_4.gif)
](images/gram01_5.gif)
| > | ip(X,Y); |
![x[1]*y[1]+x[2]*y[2]+x[3]*y[3]+x[4]*y[4]](images/gram01_6.gif){kind=small}
We'll use "nrm2(v)" for the norm squared of v.
| > | nrm2(X); |
![x[1]^2+x[2]^2+x[3]^2+x[4]^2](images/gram01_7.gif){kind=small}
and "nrm(v)" will be the norm of v:
| > | nrm(X); |
![(x[1]^2+x[2]^2+x[3]^2+x[4]^2)^(1/2)](images/gram01_8.gif){kind=small}
Example: Gram-Schmidit in 4 dimensional space.
Problem: apply the gram-schmidt process to the following 4 vectors.
| > | v1 := <1, 1, 0, 2>; v2 := <1, 0, 1, -1>; v3:=<1, 0, 0, 1>; v4 := <1, 1, 2, 1>; |
](images/gram01_9.gif)
](images/gram01_10.gif)
](images/gram01_11.gif)
](images/gram01_12.gif)
Check that these vectors form a basis:
| > | M := <v1|v2|v3|v4>; |
![M := Matrix([[1, 1, 1, 1], [1, 0, 0, 1], [0, 1, 0, 2], [2, -1, 1, 1]])](images/gram01_13.gif)
| > | Determinant(M); |
Begin the Gram-Schmidt Process:
| > | u1 := v1/nrm(v1); |
], [1/6*6^(1/2)], [0], [1/3*6^(1/2)]])](images/gram01_15.gif)
| > | nrm(v1); |
| > | u2p := v2 - ip(v2, u1)*u1; |
/3]])](images/gram01_17.gif)
| > | nrm(u2p); |
| > | u2 := u2p/nrm(u2p); |
], [1/102*102^(1/2)], [1/17*102^(1/2)], [-2/51*102^(1/2)]])](images/gram01_19.gif)
| > | u3p := v3 - ip(v3, u1)*u1 - ip(v3, u2)*u2; |
/17], [(-3)/17], [2/17]])](images/gram01_20.gif)
| > | nrm(u3p); |
| > | u3 := u3p/nrm(u3p); |
], [-9/119*119^(1/2)], [-3/119*119^(1/2)], [2/119*119^(1/2)]])](images/gram01_22.gif)
| > | u4p := v4 - ip(v4, u1)*u1 -ip(v4, u2)*u2 -ip(v4,u3)*u3; |
/7], [(-3)/7], [6/7], [3/7]])](images/gram01_23.gif)
| > | nrm(u4p); |
| > | u4 := u4p/nrm(u4p); |
], [-1/7*7^(1/2)], [2/7*7^(1/2)], [1/7*7^(1/2)]])](images/gram01_25.gif)
Check that these are really orthogonal to each other:
| > | ip(u1, u2); |
| > | ip(u1, u3); |
| > | ip(u1, u4); |
| > | ip(u2, u3); |
| > | ip(u2, u4); |
| > | ip(u3, u4); |
| > |
| > |
Modified Gram-Schmidt on the same vectors:
| > |
| > | u1p := v1; |
](images/gram01_32.gif)
| > | u2p := v2 - ip(v2, u1p)*u1p/nrm2(u1p); |
/3]])](images/gram01_33.gif)
| > | u3p := v3 - ip(v3,u1p)*u1p/nrm2(u1p) -ip(v3, u2p)*u2p/nrm2(u2p); |
/17], [(-3)/17], [2/17]])](images/gram01_34.gif)
| > | u4p := v4 - ip(v4, u1p)*u1p/nrm2(u1p)-ip(v4,u2p)*u2p/nrm2(u2p) -ip(v4, u3p)*u3p/nrm2(u3p); |
/7], [(-3)/7], [6/7], [3/7]])](images/gram01_35.gif)
| > | u1 := u1p/nrm(u1p); |
], [1/6*6^(1/2)], [0], [1/3*6^(1/2)]])](images/gram01_36.gif)
| > | u2 := u2p/nrm(u2p); |
], [1/102*102^(1/2)], [1/17*102^(1/2)], [-2/51*102^(1/2)]])](images/gram01_37.gif)
| > | u3 := u3p/nrm(u3p); |
], [-9/119*119^(1/2)], [-3/119*119^(1/2)], [2/119*119^(1/2)]])](images/gram01_38.gif)
| > | u4 := u4p/nrm(u4p); |
], [-1/7*7^(1/2)], [2/7*7^(1/2)], [1/7*7^(1/2)]])](images/gram01_39.gif)
| > |
Consider the following vector:
| > | w := <1, 2, 4, 1>; |
](images/gram01_40.gif)
| > |
Calculate the projection of w onto the span of u1 and u2:
| > | p := ip(w, u1)*u1 + ip(w, u2)*u2; |
](images/gram01_41.gif)
Calculate the matrix of the projection onto the span of u1 and u2:
| > | U := <u1|u2>; |
![U := Matrix([[1/6*6^(1/2), 7/102*102^(1/2)], [1/6*6^(1/2), 1/102*102^(1/2)], [0, 1/17*102^(1/2)], [1/3*6^(1/2), -2/51*102^(1/2)]])](images/gram01_42.gif)
| > | Pr := U.Transpose(U); |
![Pr := Matrix([[11/17, 4/17, 7/17, 1/17], [4/17, 3/17, 1/17, 5/17], [7/17, 1/17, 6/17, (-4)/17], [1/17, 5/17, (-4)/17, 14/17]])](images/gram01_43.gif)
| > |
Check that this gives the same result for the projection of w:
| > | Pr.w; |
](images/gram01_44.gif){kind=small}
Example in 3 dimensions
| > | v1 := <1, 1, 0>; v2 := < 0, 1, 1>; v3 := <1, -1, 1>; |
](images/gram01_45.gif)
](images/gram01_46.gif)
](images/gram01_47.gif)
| > | M := <v1|v2|v3>; |
![M := Matrix([[1, 0, 1], [1, 1, -1], [0, 1, 1]])](images/gram01_48.gif)
| > | Determinant(M); |
| > | u1 := v1/nrm(v1); |
], [1/2*2^(1/2)], [0]])](images/gram01_50.gif)
| > | u2p := v2 - ip(v2, u1)*u1; |
/2], [1/2], [1]])](images/gram01_51.gif)
| > | u2 := u2p/nrm(u2p); |
], [1/6*6^(1/2)], [1/3*6^(1/2)]])](images/gram01_52.gif)
| > | u3p := v3 - ip(v3, u1)*u1 - ip(v3, u2)*u2; |
](images/gram01_53.gif)
| > | u3 := u3p/nrm(u3p); |
], [-1/3*3^(1/2)], [1/3*3^(1/2)]])](images/gram01_54.gif)
| > | u1, u2, u3; |
], [1/2*2^(1/2)], [0]]), Vector[column]([[-1/6*6^(1/2)], [1/6*6^(1/2)], [1/3*6^(1/2)]]), Vector[column]([[1/3*3^(1/2)], [-1/3*3^(1/2)], [1/3*3^(1/2)]])](images/gram01_55.gif)
| > | ip(u1, u2); |
| > | ip(u1, u3); |
| > | ip(u2, u3); |
| > |
| > |
| > |
Example in the space C[0,1]
| > |
We use the inner product: 
| > | ip := proc(f,g) int(f*g, x=0..1); end; |
| > | ip(x,x); |
Consider the subspaces spanned by the following polynomials. Use Gram-Schmidt to find an ON basis of this subspace.
| > | v1 := 1; v2 := x; v3 := x^2; v4 := x^3; |
| > |
| > |
| > | u1 := v1/nrm(v1); |
| > | u2p := v2 - ip(v2,u1)*u1; |
| > | u2 := u2p/nrm(u2p); |
| > | u3p := v3 - ip(v3, u1)*u1 -ip(v3, u2)*u2; |
| > | u3 := u3p/nrm(u3p); |
| > | u4p := v4 - ip(v4,u1)*u1 - ip(v4, u2)*u2 -ip(v4,u3)*u3; |
| > | u4 := u4p/nrm(u4p); |
| > |
| > |
Same problem, modified Gram-Schmidt.
| > |
| > | u1p := v1; |
| > | u2p := v2 - ip(v2, u1p)*u1p/nrm2(u1p); |
| > | u3p := v3 - ip(v3, u1p)*u1p/nrm2(u1p) - ip(v3, u2p)*u2p/nrm2(u2p); |
| > | u4p := v4 - ip(v4, u1p)*u1p/nrm2(u1p) - ip(v4, u2p)*u2p/nrm2(u2p) - ip(v4, u3p)*u3p/nrm2(u3p); |
| > | u1 := u1p/nrm(u1p); |
| > | u2 := u2p/nrm(u2p); |
| > | u3 := u3p/nrm(u3p); |
| > | u4 := u4p/nrm(u4p); |
Plot these othogonal polynomials to see what they look like:
| > | plot([u1, u2, u3, u4], x=0..1); |
![[Plot]](images/gram01_81.gif)
Check that u3 and u4 are othogonal:
| > | Int(u3*u4,x=0..1); |
| > | value(%); |
Check that u4 is a unit vector:
| > | plot(u4^2, x=0..1); |
![[Plot]](images/gram01_84.gif)
| > | Int(u4^2, x=0..1); |
| > | value(%); |
| > |