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f[a_, b_] {2, 3, 4}.(a * b) Projection[a, b, f] 4 2 2 { 39, 13, 39}
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A useful inner product often used in function spaces is <f, g> = product, compute the projection of x2 on x3 + 1. a = x2; b = x3 + 1; f[p1_, p2_] 7 (1+ x3) 24 p1 p2 x Projection[a, b, f]
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f ( x ) g( x ) dx . Using this inner
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A finite dimensional vector space, by definition, has a finite basis. However, except for the trivial vector space that contains only the zero vector, an infinite number of different bases are possible. The most convenient basis for any vector space is an orthonormal basis. The Gram-Schmidt orthogonalization process provides a recipe for converting any basis into an orthonormal basis.
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Normalize[vector] converts vector into a unit vector. Normalize[vector, f] converts vector into a unit vector with respect to the norm function f. Orthogonalize[vectorlist] uses the Gram-Schmidt method to produce an orthonormal set of vectors whose span is vectorlist. Orthogonalize[vectorlist, f] produces an orthonormal set of vectors with respect to the inner product defined by f. n Norm[v] returns the Euclidean norm of v. || v || = vi2 .
i =1
EXAMPLE 25 To normalize (3, 4, 12) with respect to the Euclidean inner product, we type
Normalize[{3, 4, 12}] 3 4 {13, 13, 12} 13 To normalize with respect to the norm || v || = f[v_]= {2,3,4}.(v * v) Normalize[{3, 4, 12}, f]
2 2 2v12 + 3v2 + 4 v3 , we define
3 ,2 2 ,2 6 214 321 107
EXAMPLE 26 Find an orthonormal basis for the space spanned by (1, 1, 1, 0, 0), (0, 1, 1, 1, 0), and (0, 0, 1, 1, 1),
and verify that the result is correct. v = {{1, 1, 1, 0, 0}, {0, 1, 1, 1, 0}, {0, 0, 1, 1, 1}}; w = Orthogonalize[v]
, 0, 0 ,
2 15
1 15
1 15
3 1 3 ,0 , , , 5 2 10 2 10
5 2 , , 2 10 2 10 1
To verify that the result is correct, we compute six dot products. w[[1]] .w[[1]] w[[2]] .w[[2]] 1 w[[3]] .w[[3]] 1 w[[1]] .w[[2]] 0 w[[1]] .w[[3]] 0 w[[2]] .w[[3]] 0
The vectors are mutually orthogonal. The vectors are all unit length.
Linear Algebra
SOLVED PROBLEMS
12.30 Compute the norm of the vector (1, 2, 3, 4, 5) with respect to (a) the Euclidean inner product and (b) <u, v> = 2u1v1 + 3u2v2 + u3v3 + 3u4v4 + 2u5v5.
SOLUTION
(a) u = {1,2,3,4,5}; norm = 55 u.u
(b) u = {1,2,3,4,5}; c = {2,3,1,3,2}; norm = 11 c.(u * u)
12.31 Find the projection of the vector (3, 4, 5) onto each of the coordinate axes.
SOLUTION
v = {3, 4, 5}; Projection[v, {1, 0, 0}]
0, 0} 4, 0} 0, 5}
Projection[v, {0, 1, 0}]
Projection[v, {0, 0, 1}]
12.32 Find a unit vector having the same direction as (1, 2, 2, 3).
SOLUTION
Normalize[{1, 2, 2, 3}] 1 , 2, 2, 1 3 3 2 3 2
12.33 If a = (1, 2, 3) and b = (1, 2, 5), compute the length of the vector v shown in the diagram.
b v SOLUTION a
Since b + v = projab, it follows that v = projab b. a = {1, 2, 3}; b = {1, 2, 5}; v = Projection[b, a] b; Norm[v] 138 7
12.34 Find an orthonormal basis for the space spanned by (1, 2, 1, 3), (2, 2, 2, 2), (1, 1, 1, 1), and (3, 4, 3, 5).
SOLUTION
v1 = {1, 2, 1, 3}; v2 = {2, 2, 2, 2};
Linear Algebra
v3 = {1, 1, 1, 1}; v4 = {3, 4, 3, 5}; v = {v1, v2, v3, v4}; w = Orthogonalize[v]
1 2 , , 15 15 1 , 2 22
1 8 3 1 3 8 , , , , 2 , , 55 15 5 165 165 165 1 2 2 The set v is linearly dependent, so Orthogonalize , , , {0,0,0,0} can only produce three basis vectors. (0, 0, 0, 0} 11 22 11 can be disregarded.
12.35 Construct an orthonormal basis for P5, the set of all polynomials of degree 5 with respect to the 1 inner product <p, q> = p( x )q( x )dx.
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