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w + 2 x + 3 y + 3z = 9 3w + 4 x + 4 y + 5 z = 16 12.28 Find the general solution of the system 2w + 2 x + y + 2z = 7 4 w + 6 x + 7 y + 8 z = 25
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SOLUTION
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1 3 a = 2 4
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b ={9,16,7,25};
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Det[a] 0 True nullspacebasis = NullSpace[a] {{1, 2, 0, 1}, {4, 5, 2, 0}} particular = LinearSolve[a, b]
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{ 2, 11,0,0} 2
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generalsolution = s * nullspacebasis[[1]]+ t * nullspacebasis[[2]] + particular
{ 2 + s + 4 t, 11 2 s 5 t,2 t,s} 2
12.29 Let A be a 7 7 tridiagonal matrix having 3s on the main diagonal and 1s on the diagonals adjacent to the main diagonal. Let ei be a 7-dimensional vector having 1 in the ith position and 0s elsewhere. Solve Ax = ei , i = 1, . . . , 7.
SOLUTION
a = Table[If[Abs[i j] 1, 1, If[i j, 3, 0]], {i, 1, 7}, {j, 1, 7}]; a // MatrixForm 3 1 0 0 0 0 0 1 3 1 0 0 0 0 0 1 3 1 0 0 0 0 0 1 3 1 0 0 0 0 0 1 3 1 0 0 0 0 0 1 3 1 0 0 0 0 0 1 3 ludata = LUDecomposition[a]; b = Table[KroneckerDelta[i, j], {i, 1, 7}, {j, 1, 7}]; LUBackSubstitution[ludata, b] // TableForm
377 987 48 329 55 987 1 47 8 987 1 329 1 987 48 329 144 329 55 329 3 47 8 329 3 329 1 329 55 987 55 5 329 440 987 8 47 64 987 8 8 329 8 987 1 47 3 47 8 47 21 47 8 47 3 47 1 47 8 987 8 329 64 987 7 8 47 440 987 55 329 55 987 1 329 3 329 8 329 3 47 55 329 4 144 329 48 329 1 987 1 329 8 987 1 47 55 987 48 329 377 987
KroneckerDelta[i, j]= 1 if i = j and 0 otherwise.
The ith column of the table represents the solution of Ax = ei.
Linear Algebra
12.5 Orthogonality
Two vectors are orthogonal if their inner product is 0. Orthogonal vectors possess useful properties that make working with them convenient. For example, if u and v are orthogonal, they satisfy the (generalized) Theorem of Pythagoras: || u + v ||2 = || u ||2 + || v ||2. Orthogonality also allows us to introduce the concept of projection. In 2 it is easy to visualize what projection means. If a = PQ and b = PR are two vectors with the same initial point P, then if S is the foot of the perpendicular from R to PQ, the projection of b onto a is
Q S P
the vector PS . This vector is often represented as projab. The projection vector can be computed using the Mathematica command Projection.
Projection[vector1, vector2] returns the orthogonal projection of vector1 onto vector2.
EXAMPLE 22 Compute the projection of (1, 2, 3) onto ( 2, 3, 1).
a = {1, 2, 3}; b = { 2, 3, 1}; Projection[a, b] 3 1 { 1, 14, 14} 7
The concept of orthogonality depends upon the definition of inner product for the space under consideration. By default, Mathematica uses the Euclidean inner product (dot product) in linear algebra commands. However, this can be changed by including an alternate definition in the third argument of Projection.
Projection[vector1, vector2, f ] returns the orthogonal projection of vector1 onto vector2 with respect to an inner product defined by f.
It can be shown that if c1, c2, and c3 are positive real numbers, then <a, b> = c1 a1 b1 + c2 a2 b2 + c3 a3 b3 defines an inner product on 3. To compute the orthogonal projection of (1, 2, 3) onto ( 2, 3, 1) using this inner product, we must define an appropriate function describing the inner product. To do this, we compute a * b and then take the dot product with c = (c1, c2, c3).
EXAMPLE 23 Compute the projection of (1, 2, 3) onto ( 2, 3, 1) using the inner product <a, b> = 2a1b1 + 3a2b2 + 4a3b3.
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