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Elementary Linear Algebra with Applications 9th edition Howard Anton, Chris Rorres - Solutions

Find the Fourier series of f(x) = 1, 0 < x < π and f(x) = 0, π ≤ x ≤ 2π over the interval [0, 2π].
In each part, classify the quadratic form as positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite. (a) x21 + x22 (b) (x1 - x2)2 (c) x21 - x22
Let xTAx be a quadratic form in x1, x2,..., xn and define T: Rn → R by T(x) = xTAx. (a) Show that T(x + y) = T(x) + 2xTAy + T(y). (b) Show that T = (kx) = k2T(x). (c) Is T a linear transformation? Explain.
Express the quadratic form (c1x1 + c2x2 + ... + cnxn)2 in the matrix notation xTAx, where A is symmetric.
Complete the proof of Theorem 9.5.1 by showing that λn ≤ xTAx if ||x|| = 1 and λn = xTAx if x is an eigenvector of A corresponding to λn.
In each part, find the maximum and minimum values of the quadratic form subject to the constraint x21 + x22 = 1, and determine the values of x1 and x2 at which the maximum and minimum occur. 7x21 + 4x22 + x1x2
Use Theorem 9.5.2 to determine which of the following matrices are positive definite.
Use Theorem 9.5.2 to determine which of the following matrices are positive definite.
In each part, find a change of variables that reduces the quadratic form to a sum or difference of squares, and express the quadratic form in terms of the new variables. 2x21 + 2x22 - 2x1x2
The graph of a quadratic equation in x and y can, in certain cases, be a point, a line, or a pair of lines. These are called degenerate conics. It is also possible that the equation is not satisfied by any real values of x and y. In such cases the equation has no graph; it is said to represent an
In each part, a translation will put the conic in standard position. Name the conic and give its equation in the translated 7 coordinate system. (a) 9x2 + 4y2 - 36x - 24y + 36 = 0 (b) y2 - 8x - 14y + 49 = 0 (c) 2x2 - 3y2 + 6x + 20y = -41
Prove Theorem 9.7.1. Theorem 9.7.1 Principal Axes Theorem for ft3 Let ax2 + by2 + cz2 + 2dxy + 2exz + 2fyz + gx + hy + iz + j = 0 be the equation of a quadric Q, and let xTAx = ax2 + by2 + cz2 + 2dxy + 2exz + 2fyz be the associated quadratic form. The coordinate axes can be rotated so that the
In Exercise 4, identify the trace in the plane z = l in each case. Exercise 4 Name the following quadrics. (a) 36x2 + 9y2 + 4z2 - 36 = 0 (b) 2x2 + 6y2 - 3z2 = 18 (c) 6x2 - 3y2 - 2z2 - 6 = 0 (d) 9x2 + 4y2 - z2 = 0 (e) 16x2 + y2 = l6z (f) 7x2 - 3y2 + z = 0 (g) x2 + y2 + z2 = 25
In each part, determine the translation equations that will put the quadric in standard position, and find the equation of the quadric in the translated coordinate system. Name the quadric. (a) 9x2 + 36y2 + 4z2 - 18x - 144y - 24z + 153 = 0 (b) 3x2 - 3y2 - z2 + 42x + 144 = 0 (c) x2 + 16y2 + 2x - 32y
Find the number of additions and multiplications required to compute AB if A is an m × n matrix and B is an m × p matrix.
Consider the variation of Gauss-Jordan elimination in which zeros are introduced above and below a leading 1 as soon as it is obtained, and let A be an invertible n × n matrix. Show that to solve a linear system Ax = b using this version of Gauss-Jordan elimination requires n3/2 + n2/2
Derive the formula 1 + 2 + 3 + ... + n = n(n + 1)/2 Let Sn = 1 + 2 + 3 + ... + n. Write the terms of Sn in reverse order and add the two expressions for Sn.
Derive the formula l2 + 22 + 32 + ... + n2 = n(n + 1) (2n +1)/6 Using the following steps. (a) Show that (k + l)3 - k3 = 3k2 + 3k + 1. (b) Show that [23 - 13] + [33 - 23] + [43 - 33] + ... + [(n + 1)3 - n3] = (n + 1)3 - 1 (c) Apply (a) to each term on the left side of (b) to show that (n + l)3 - l
Let R be a row-echelon form of an invertible n × n matrix. Show that solving the linear system Rx = b by back-substitution requires n2/2 - n/2 multiplications and n2/2 - n/2 additions
Use the method of Example 1 and the LU-decompositionTo solve the system 3x1 - 6x2 = 0 -2x1 + 5x2 = 1
Let(a) Find an LU-decomposition of A. (b) Express A in the form A = L1DU1, where L1 is lower triangular with l's along the main diagonal, U1 is upper triangular, and D is a diagonal matrix. (c) Express A in the form A = L2U2, where L2 is lower triangular with l's along the main diagonal and U2
Let(a) Prove: If a ‰ 0, then A has a unique LU-decomposition with l's along the main diagonal of L. (b) Find the LU-decomposition described in part (a).
Recall from Theorem 1.7.1 b that a product of lower triangular matrices is lower triangular. Use this fact to prove that the matrix L in 8 is lower triangular.
Prove: If A is any n × n matrix, then A can be factored as A = PLU, where L is lower triangular, U is upper triangular, and P can be obtained by interchanging the rows of ln appropriately.
Show that if A = PLU, then Ax = b may be solved by a two-step process similar to the process in Example 1. Use this method to solve Ax = b, where A is the matrix in Exercise 18 and b = e2.Matrix in Exercise 18
Find an LU-decomposition of the coefficient matrix; then use the method of Example 1 to solve the system.a.b.
Show that Im (iz) = Re (z)
(a) Show that if n is a positive integer, then the only possible values for in are 1, - 1, i, and (b) Find i2509.
Use the result of Exercise 24 to prove: If zz1 = zz2 and z ≠ 0, then z1 = z2. Exercise 24: Prove: If z1z2 = 0, then z1 = 0 or z2 = 0.
Prove that for all complex numbers z1, z2, and z3, z1z2 = z2z1
In each part, use the given information to find the real numbers x and y. (x + y) + (x - y)i = 3 + i
In each part, solve for z. (a) z + (1 - i ) = 3 + 2i (b) (i - z) + (2z - 3i) = - 2 + 7i
In each part, find z1z2, z12, and z22 (a) z1 = 3i, z2 = 1 - i (b) z1 = 4 + 6i, z2 = 2 - 3i (c) z1 = 1/3 (2 + 4i), z2 = 1/2 (1 - 5i)
In each part, solve for z. iz = 2 - i
In each part, sketch the set of points in the complex plane that satisfies the equation. (a) |z| = 2 (b) |z - i| = |z + i|
Given that z = x + iy, find (a) Re () (b) Re ()
(a) Prove that z = ()2. (b) Prove that if n is a positive integer, then z = ()n.
Verify that |z= |z|2| of for (a) z = 2 - 4i (b) z = - 3 + 5i
In each part, use the formula in Theorem 1.4.5 to compute the inverse of the matrix, and check your result by showing that AA-1 = A-1 A = I.(a)
In each part, use the method of Example 4 in Section 1.5 to find and check your result by showing that AA-1 = A-1 A = I.(a)
(a) If z1 = a1 + b1i and z2 = a2 + b2i, find |z1 - z2| interpret the result geometrically. (b) Use part (a) to show that the complex numbers 12, [6 + 2i, and 8 + 8i are vertices of a right triangle.
In each part, find l / z. (a) z = i (B) z = - i/7
In each part, find the principal argument of z. (a) z = l (b) z = - i (c) z = - 1 + √3i
In each part, find Re (z) and Im (z). (a) z = 3eiπ (b) |= √2eπi/2|
Show that Formula 7 is valid if n = 0 or n is a negative integer.
Use Formula 11 to show that
Show that Formula 6 is valid for negative integer exponents if z ≠ 0.
In each part, express the complex number in polar form using its principal argument. (a) 2i (b) 5 + 5i (c) - 3 - 3i
Express z1 = i, z2 = 1 - √3i, and z3 = √3 + i in polar form, and use your results to find z1z2/z3. Check your results by performing the calculations without using polar forms.
In each part, find all the roots and sketch them as vectors in the complex plane. (a) (- i)1/2 (b) (- 27)1/3
Use the method of Example 4 to find all sixth roots of 1.
Let u = (2i, 0, - 1,(3), v = (- i, i, 1 + i, - 1) and w = (1 + i, - i, - 1 + 2i, 0). Find (a) u - v (b) - w + v (c) - iv + 2iw
A set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are complex vector spaces under the given operations. For those that are not, list all axioms that fail to hold.The set of all complex 2 Ã— 2 matrices of the formwith the
Let T: C3 †’ C3 be a linear operator defined by T(x) = Ax, whereFind the kernel and nullity of T.
Use Theorem 5.2.1 to determine which of the following are subspaces of the vector space of complex-valued functions of the real variable x:(a) All f such that f (1) = 0
Express the following as linear combinations of u = (1, 0, - i), v = (1 + i, 1, 1 - 2i), and w = (0, i, 2). (a) (1, 1, 1) (b) (0, 0, 0)
Determine which of the following lie in the space spanned by f = eix and g = e-ix (a) cos x (b) sin x (c) cos x + 3i sin x
Which of the following sets of vectors in C3 are linearly independent? (a) u1 = (1 - i, 1, 0), u2 = (2, 1 + i, 0), u3 = (1 + i, i, 0) (b) u1 = (z, 0, 2 - i), u2 = (0, 1, i), u3 = (- i, - 1 - 4i, 3)
Which of the following sets of vectors are bases for (a) (2i, - i), (4i, 0) (b) (2 - 3i, i), (3 + 2i, - 1)
Determine the dimension of and a basis for the solution space of the system. x1 + (1 + i)x2 = 0 (1 - i)x1 + 2x2 = 0
Let u1 = (1 - i, i, 0), u2 = (2i, 1 + i, 1), and u3 = (0, 2i, 2 - i). Find scalars c1, c2 and c3 such that c1u1 + c2u2 + c3u3 = (- 3 + i, 3 + 2i, 3 - 4i).
Prove: If u and v are vectors in complex Euclidean w-space, then u ∙ (kv) = (u ∙ v)
Establish the identityfor vectors in complex Euclidean n-space.
Find the Euclidean norm of v if (a) v = (1, i) (b) v = (2i, 0, 2i + 1, - 1)
Find the Euclidean inner product u ∙ v if. (a) u = ( - i 3i), v = (3i, 2i) (b) u = (3 - 4z, 2 + i, - 6i),v = (1 + i, 2 - i, 4)
Let u = (u1, u2) and v = (v1, v2). Show that (u, v) = 3u1I + 2u22 defines an inner product on C2.
Use the inner product of Exercise 3 to find ||w|| if (a) w = (1, - i) (b) w = (3 - 4i, 0)
Let C2 have the inner product of Exercise 1. Find d(x, y) if x = (1, 1), y = (i, - i)
Repeat the directions of Exercise 13 using the inner product of Exercise 3. x = (1, 1), y = (i, - i)
Let C3 have the Euclidean inner product. For which complex values of k are u and v orthogonal? u = (2i, i, 3i), v = (i, 6i, k)
Let C3 have the Euclidean inner product. Show that for all values of the variable Î¸, the vectorhas norm 1 and is orthogonal to both (1, i, 0) and (0, i, - i).
Let C3 have the Euclidean inner product. Which of the following form orthonormal sets?
Let C3 have the Euclidean inner product. Else the Gram-Schmidt process to transform the basis (u1, u2, u3) into an orthonormal basis. u1 = (i, i, i), u2 = (- i, i, 0), u3 = (i, 2i, i)
Let C3 have the Euclidean inner product. Find an orthonormal basis for the subspace spanned by (0, i, 1 - i) and (- i, 0, 1 + i).
Prove If k is a complex number and (u, v) is an inner product on a complex vector space, then Œ©u - kv, u - kvŒª = Œ©u, uŒª - (u, v) - + (v, v)
Let u = (u1, u2) and v = (v1, v2). Show that (u, v) = u11 + (1 + i)u12 + (1 - i)u21 + 3u22 defines an inner product on C2.
Prove that if u and v are vectors in a complex inner product space, then
Prove that if f = f1(x) + i f2(x) and g = g1(x) + ig2(x) are vectors in complex C[a, b] then the formuladefines a complex inner product on C[a, b].
Let f = f1(x) + i f2(x) and g = g1(x) + ig2(x) be vectors in complex C[0, 1] and let this space have the inner product defined in Exercise 39. Show that the vectors where m = 0, 11, 2, ..., form an orthonormal set.
Let u = (u1, u2) and v = (v1, v2). Determine which of the following are inner products on For those that are not, list the axioms that do not hold. (a) (u, v) = u11 (b) (u, v) = |u1|2|v1|2 + |u2|2|v2|2 (c) (u, v) = 2u11 + iu12 - iu21 + 2u22
Let C2 have the inner product of Exercise 1. Lind ||w|| if (a) w = (- i, 3i) (b) w = (0, 2 - i)
Show that the eigenvalues of the symmetric matrixare not real. Does this violate Theorem 10.6.6?
Prove: If A is an n x n matrix with complex entries, then
Prove: If A is invertible, then so is A*, in which case (A*)-1 = (A-1)*.
Prove that an n × n matrix with complex entries is unitary if and only if its rows form an orthonormal set in Cn with the Euclidean inner product.
Let λ and μ be distinct eigenvalues of a Hermitian matrix A. (a) Prove that if x is an eigenvector corresponding to λ and y an eigenvector corresponding to μ, then x*Ay = λx*y and x* Ay = μx* y. (b) Prove Theorem 10.6.4.
Find a unitary matrix p that diagonalizes A, and determine P-1 AP.7.9. 11.
Show that the eigenvalues of a unitary matrix have modulus 1.
Find a basis for the solution space of the system
Find the eigenvalues of the matrixwhere Ï‰ = e2Ï€i/3.
Show that if U is an n Ã— n unitary matrix and |z1| = |z2| = ... = |zn| = 1, then the productis also unitary.
Find the equations of the lines that pass through the following points: (a) (1, - 1), (2, 2) (b) (0, 1), (1, - 1)
Find a determinant equation for the parabola of the form c1y + c2x2 + c3x + c4 = 0. that passes through three given noncollinear points in the plane.
Find the equation of the conic section that passes through the points (0, 0), (0, - 1), (2, 0), (2, - 5), and (4, - 1).
Show that Equation 10 is the equation of the conic section that passes through five given distinct points in the plane.
A certain forest is divided into three height classes and has a growth matrix between harvests given byIf the price of trees in the second class is $30 and the price of trees in the third class is $50, which class should be completely harvested to attain the optimal sustainable yield? What is the
In Example 1, what must the ratio of the prices p2: p3: p4: p5: p6 be in order that the yields Yldk, k = 2, 3, 4, 5, 6, all be the same? (In this case, any sustainable harvesting policy will produce the same optimal sustainable yield.)
For the optimal sustainable harvesting policy described in Theorem 11.10.2, how many trees are removed from the forest during each harvest?
Find the currents in the circuits.1.3.
Show that if the current in the I5 circuit of the accompanying figure is zero, then R4 = R3R2/R1.

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