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m[[i,i]]
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12.21 Construct a 9 9 block diagonal matrix with a 2 2 block of 2s, a 3 3 block of 3s and a 4 4 block of 4s. (A block diagonal matrix is a square partitioned matrix whose diagonal matrices are square and all others are zero matrices.)
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SOLUTION
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m2 = Table[2, {2}, {2}]; m3 = Table[3, {3}, {3}];
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Linear Algebra
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m4 = Table[4, {4}, {4}]; z27 = ConstantArray[0, {2, 7}]; z32 = ConstantArray[0, {3, 2}]; z34 = ConstantArray[0, {3, 4}]; z45 = ConstantArray[0, {4, 5}]; top = ArrayFlatten[{{m2, z27}}]; middle = ArrayFlatten[{{z32, m3, z34}}]; bottom = ArrayFlatten[{{z45, m4}}]; ArrayFlatten[{{top}, {middle}, {bottom}}] //MatrixForm 2 2 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 3 3 3 0 0 0 0 0 0 3 3 3 0 0 0 0 0 0 3 3 3 0 0 0 0 0 0 0 0 0 4 4 4 4 0 0 0 0 0 4 4 4 4 0 0 0 0 0 4 4 4 4 0 0 0 0 0 4 4 4 4
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z27 is a 2 7 array of zeros, etc.
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a f 12.22 Let M = k p u
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Find the matrix P obtained from M by deleting its fourth row and third column.
SOLUTION
Generates a list of alphabet letters. Forms five sublists of five letters each.
temp = CharacterRange["a", "y"]; m = Partition[temp, 5]; m // MatrixForm a f k p u c d e g h i j l m n o q r s t v w x y b
Drop[m,{4},{3}] //MatrixForm a f k u b d e g i j l n o v x y
Linear Algebra
12.4 Linear Systems of Equations
Mathematica offers a number of ways to solve systems of linear equations. Solve, discussed in 6, offers one alternative, but it is somewhat clumsy and inefficient for use on large systems. In this section we discuss a number of other procedures for solving systems of linear equations.
LinearSolve[a, b] produces vectors, x, such that a.x = b. LinearSolve[a] produces a LinearSolveFunction that can be used to solve a.x = b for different vectors b.
Here a is the matrix of coefficients of the unknowns, and b is the right-hand side of the linear system. If a is invertible, LinearSolve will produce a unique solution to the linear system. If a is singular, either no solution exists or there are an infinite number of solutions. If a system has a unique solution, Mathematica returns the solution. If no solution exists, Mathematica returns an error message.
EXAMPLE 15 The system 2 x + y + z = 7, x 4 y + 3 z = 2, 3 x + 2 y + 2 z = 13 has a unique solution.
The system 2 x + y + z = 7, x 4 y + 3 z = 2, 3 x 3 y + 4 z = 13 has no solution. 2 1 1 a1 = 1 4 3 ; 3 2 2 2 1 1 a2 = 1 4 3 ; 3 3 4 b ={7,2,13}; LinearSolve[a1, b] {1, 2, 3} LinearSolve[a2, b] LinearSolve nosol : Linear equation encountered that has no solution. LinearSolve[{{2, 1, 1}, {1, 4, 3}, {3, 3, 4}}, {{7}, {2}, {13}}]
If the system a.x = b has an infinite number of solutions, the treatment is a bit more complicated. In this case, Mathematica returns one solution, known as a particular solution. The full set of solutions is constructed by adding to the particular solution the set of all solutions of the corresponding homogeneous system, a.x = 0. The set of all vectors, x, such that a.x = 0, is called the null space of a and is easily determined by the command NullSpace.
NullSpace[a] returns the basis vectors of the null space of a.
The nullity of a, the dimension of the null space of a, can be found by computing Length[NullSpace[a]]. The rank of a may be computed as n Length[NullSpace[a]] where n represents the number of columns of a.
EXAMPLE 16 2 x + y + z = 7, x 4 y + 3 z = 2, 3 x 3 y + 4 z = 9 has an infinite number of solutions.
2 1 1 a = 1 4 3 ; 3 3 4 {{ 7, 5, 9}}
b ={7,2,9};
nullspacebasis = NullSpace[a] particular = LinearSolve[a, b]
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