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Questions and Answers of Linear Algebra

In each case, show that T is an isometry of R3, determine the type (Theorem 6), and find the axis of any rotations and the fixed plane of any reflections involved.(a)(b)(c)(d)(e)(f)
If T is an isometry, show that aT is an isometry if and only if a = ±1.
In each case, find the Fourier approximation f3 of the given function in C[-Ï€, Ï€](a) f(x) = Ï€ - x(b)(c) f(x) = x2(d)
Show that {1, cosx, cos(2x), cos(3x),...} is an orthogonal set in C[0, π] with respect to the inner product (f, g) = ∫n0 f(x)g(x)dx.
In each case, find a matrix P such that P-1AP is in block triangular form as in Theorem 1.(a)(b) (c)
If T: V → V is a linear operator where V is finite dimensional, show that cT(T) = 0.
Show thatis similar to
Solve each of die following for the real number x. (a) (2 + xi)(3 - 2i) = 12 + 5i (b) (2 + xi)(2 - xi) = 5
In each case, show that u is a root of the quadratic equation, and find the other root. (a) x2 + ix - (4 - 2i) = 0; u = - 2 (b) x2 + 3(1 - i)x - 5i = 0; u = - 2 + i
Find the roots of each of the following complex quadratic equations. (a) x2 - + (1 - i) = 0 (b) x2 - 3(1 - i)x - 5i = 0
In each case, describe the graph of the equation (where z denotes a complex number). (a) |z - 1| = 2 (b) z = - (c) im z = m re z, m a real number
Express each of the following in polar form (use the principal argument). (a) - 4i (b) - 4 + 4√3i (c) - 6 + 6i
Express each of the following in the form a + bi. (a) e7πi/3 (b) √2e-πi/4 (c) 2√3e-2πi/6
Convert each of the following to the form a + bi. (a) (3 - 2i)(1 + i) + |3 + 4i| (b) 3 - 2i / 1 - i - 3 - 7i / 2 - 3i (c) (2 - i)3 (d) (1 - i)2 (2 + i)2
Express each of the following in the form a + bi. (a) (1 + √3i)-4 (b) (1 - i)10 (c) (√3 - i)9(2 - 2i)5
Find all complex numbers z such that: (a) z4 = 2/(√3i - 1) (b) z6 = - 64
(a) Suppose z1, z2, z3, z4, and z5 are equally spaced around the unit circle. Show that z1 + z2 + z3 + z4 + z5 = 0 (b) Repeat (a) for any n ≥ 2 points equally spaced around the unit circle. (c) If
In each case, find the complex number z. (a) (i + z) - 3i(2 - z) = iz +1 (b) z2 = 3 - 4i (c) z(2 - i) = ( + 1)(1 + i)
In each case, find the roots of the real quadratic equation. (a) x2 - x + 1 = 0 (b) 2x2 - 5x + 2 = 0
Find all numbers x in each case. (a) x3 = - 8 (b) x4 = 64
In each case, find a real quadratic with it as a root, and find the other root. (a) u = 2 - 3i (b) u = 3 - 4i
Find a real polynomial of degree 4 with 2 - i and 3 - 2i as roots.
In each case prove the result and either prove the converse or give a counterexample. (a) If m is an even integer and n is an odd integer, then m + n is odd. (b) If x2 - 5x + 6 = 0, then x = 2 or x =
In each case either prove the result by splitting into cases, or give a counterexample. (a) If n is any integer, then n2 = 4k + 1 for some integer k. (b) If n is any odd integer, then n2 = 4k + 1 for
In each case prove the result by contradiction and either prove the converse or give a counterexample. (a) If n + m = 25 where n and m are integers, then one of n and m is greater than 12. (b) If m
Prove each implication by contradiction, (a) If x and y are positive numbers, then √x + y ≠ √x + √y. (b) If x is irrational and y is rational, then x + y is irrational. (c) If 13 people are
Disprove each statement by giving a counterexample.(a) n2 + n + 11 is a prime for all positive integers n.(b) n3 ‰¥ 2n for all integers n ‰¥ 2.(c) If n ‰¥ 2 points are
1 / √1 + 1 / √2 + ... + 1 / √n ≤ 2 √n - 1 Prove the given statement by induction for all n ≥ 1.
n3 - n is a multiple of 3. Prove the given statement by induction for all n ≥ 1.
Let Bn = 1 ∙ 1! + 2 ∙ 2! + 3 ∙ 3! + ... + n ∙ n! Find a formula for Bn and prove it.
Suppose S" is a statement about n for each n ≥ 1. Explain what must be done to prove that Sn is true for all n ≥ 1 if it is known that (a) Sn ⇒ Sn+2 for each n ≥ 1. (b) Sn ⇒ Sn+8 for each n
1 / 1 ∙ 2 + 1 / 2 ∙ 3 + ... + 1 / n(n + 1) = n / n + 1 Prove the given statement by induction for all n ≥ 1.
Given the linear system 3x + 4y = s 6x + Sy = t, (a) Determine particular values for s and t so that the system is consistent. (b) Determine particular values for 51 and t so that the system is
Show that the linear system obtained by interchanging two equations in (2) is equivalent to (2).
Show that the linear system obtained by adding a multiple of an equation in (2) to another equation is equivalent to (2).
Describe the number of points that simultaneously lie in each of the three planes shown in each part of Figure 1.2.
Let C1 and C2 be circles in the plane. Describe the number of possible points of intersection of C1 and C2. Illustrate each case with a figure.
Suppose that the three points (1, - 5), (- 1, 1), and (2, 7) lie on the parabola p(x) = ax2 + bx + c. (a) Determine a linear system of three equations in three unknowns that must be solved to find a,
If x is an n-vector, show that x + 0 = x.
Identify the following expressions as true or false. If true, prove the result; if false, give a counterexample.(a)(b) (c)
Determine the incidence matrix associated with each of the following graphs:(a)(b)
If possible, compute the indicated linear combination:(a) C + E and E + C(b) A + B(c) D - F(d) -3C + 5O(e) 2C - 3E(f) 2B + FLet
If possible, compute the following: (a) AT and (AT)T (b) (C + E)T and CT + ET (c) (2D + 3F)T (d) D - DT (e) 2AT + B (f) (3D-2F)T
If possible, compute the following: (a) DA + B (b) EC (c) CE (d) EB + E (e) FC + D
If possible, compute the following: (a) A(BD) (b) (AB)D (c) A(C + E) (d) AC + AE (e) (2AB)T and 2(AB)T (f) A (C - 3E)
Let A = [1 2 - 3], B = [- 1 4 2], and C = [- 3 0 1]. If possible, compute the following: (a) ABT (b) CAT (c) (BAT)C (d) ATB (e) CCT (f) CTC (g) BTCAAT
(a) Let A be an m × n matrix with a row consisting entirely of zeros. Show that if B is an n × p matrix, then A B has a row of zeros. (b) Let A be an m x n matrix with a column consisting entirely
Consider the following linear system: 3x1 + 3x2 - 3x3 + x4 + x5 = 7 3x1 + 2x3 + 3x5 = - 2 2x1 + 3x2 - 4x4 = 3 x3 + x4 + x5 = 5. (a) Find the coefficient matrix. (b) Write the linear system in matrix
Write the linear system whose augmented matrix is(a)(b)
Write each of the following linear systems as a linear combination of the columns of the coefficient matrix: (a) 3x1 + 2x2 + x3 = 4 x1 - x2 + 4x3 = -2 (b) -x1 + x2 = 3 2x1 - x2 = -2 3x1 + x2 =1
Write each of the following as a linear system in matrix form:(a)(b)
If A = [ajj] is an n Ã— n matrix, then the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A,Show each of the following: (a) Tr(cA) = c Tr(A), where c is
(a) Show that the jth column of the matrix product A B is equal to the matrix product Abj, where bj is the jth column of B. It follows that the product AB can be written in terms of columns asAB =
Show that the 7th column of the matrix product AS is a linear combination of the columns of A with coefficients the entries in bj, the jth column of B.
A diet research project includes adults and children of both sexes. The composition of the participants in the project is given by the matrixThe number of daily grams of protein, fat, and
Let a, b, and c be n-vectors and let k be a real number, (a) Show that a ∙ b = b ∙ a. (b) Show that (a + b) ∙ c = a ∙ c + b ∙ c. (c) Show that (ka) ∙ b = a ∙ (kb) = k(a ∙ b).
Let A be an m × n matrix whose entires are real numbers. Show that if AAT = O (the m × m matrix all of whose enteries are zero), then A = O.
Find three 2 × 2 matrices, A, B, and C such that AB - AC with B ≠ C and A ≠ O.
Determine all 2 × 2 matrices A such that AB = BA for any 2 × 2 matrix B.
Show that (A - B)T = AT - BT.
Let x1 and x2 be solutions to the homogeneous linear system Ax = 0. (a) Show that x1 + x2 is a solution. (b) Show that x1 - x2 is a solution. (c) For any scalar r, show that rx1 is a solution. (d)
Let A = [aij] be the n × n matrix defined by aij = k and aij = 0 if i ≠ j. Show that if B is any n x n matrix, then AB = kB.
Let A be an m Ã— n matrix and C = [c1 c2 ˆ™ ˆ™ ˆ™ cm] a 1 Ã— m matrix. Prove thatWhere Aj is the jth row of A.
Show that if A is any m × n matrix, then Im A = A and AIn = A.
If p is a nonnegative integer and c is a scalar, show that (cA)P = cpAp.
For a square matrix A and a nonnegative integer p, show that (AT)P = (AP)T.
For a nonsingular matrix A and a nonnegative integer p, show that (AP)-1 = (A-1)P.
For a nonsingular matrix A and nonzero scalar k, show that (kA)-1 = 1/kA-1.
(a) Show that every scalar matrix is symmetric. (b) Is every scalar matrix nonsingular? Explain. (c) Is every diagonal matrix a scalar matrix? Explain.
(a) Show that A is symmetric if and only if aij = aij, for all i, j. (b) Show that A is skew symmetric if and only if aji = -ajj for all i, j. (c) Show that if A is skew symmetric, then the elements
Prove that the sum, product, and scalar multiple of diagonal, scalar, and upper (lower) triangular matrices is diagonal, scalar, and upper (lower) triangular, respectively.
Show that if A is any n × n matrix, then (a) A + AT is symmetric. (b) A - AT is skew symmetric.
Let A and B be symmetric matrices. (a) Show that A + B is symmetric. (b) Show that AB is symmetric if and only if AB = BA.
(a) Show that if A is an upper triangular matrix, then AT is lower triangular. (b) Show that if A is a lower triangular matrix, then AT is upper triangular.
Show that if A is an n × n matrix, then A = S + K, where S is symmetric and K is skew symmetric. Also show that this decomposition is unique.
Prove: If A and B are n × n diagonal matrices, then AB = BA.
Prove that if one row (column) of the n × n matrix A consists entirely of zeros, then A is singular.
Prove: if A is a diagonal matrix with nonzero diagonal entries a11,a22, . . . , ann, then A is nonsingular and A-1 is a diagonal matrix with diagonal entries 1/a11, 1/a22, . . . 1/ann.
Prove that if A is symmetric and nonsingular, then A-1 is symmetric.
Let A and B be the following matrix:And Find AB by partitioning A and B in two different ways.
What type of matrix is a linear combination of symmetric matrices? Justify your answer.
What type of matrix is a linear combination of scalar matrices? Justify your answer.
Determine the command for computing the inverse of a matrix in the software you use. Usually, if such a command is applied to a singular matrix, a warning message is displayed. Experiment with your
In Section 1.1 we studied the method of elimination for solving linear systems Ax = b. In Equation (2) of this section we showed that the solution is given by x = A-l b, if A is nonsingular. Using
Experiment with your software to determine the behavior of the matrix sequence Ak as k †’ ˆž for each of the following matrices:(a)(b)
Let p and q be nonnegative integers and let A be a square matrix. Show that Ap Aq = Ap+q and (Ap)q = Apq.
If AB = BA and p is a nonnegative integer, show that (AB)P = APBP.
F: R2 †’ R2 (reflection with respect to the y-axis) defined bySketch u and its image under each given matrix transformation f.
Let f: Rn → Rm be a matrix transformation defined by f(u) = Au, where A is an m x n matrix. (a) Show that f(u + v) = f(u) + f(v) for any u and v in Rn. (b) Show that f(cu) = cf(u) for any u in Rn
Let f: Rn Rm be a matrix transformation defined by f(u) = Au, where A is an m x n matrix. Show that if u and v are vectors in Rn such that f(u) = 0 and f(v) = 0, whereThen f(cu + dv) = 0 for any real
(a) Let O: Rn → Rm be the matrix transformation defined by 0(u) = Ou, where O is the m × n zero matrix. Show that O(u) = 0 for all u in Rn. (b) Let I: Rn → Rn be the matrix transformation
F: R2 †’ R2 is a counterclockwise rotation through 2 / 3 Ï€ radians;Sketch u and its image under each given matrix transformation f.
Sketch u and its image under each given matrix transformation f.F: R2 †’ R2 defined by
Sketch u and its image under each given matrix transformation f.F: R2 †’ R2 defined by
Let f be the matrix transformation defined in Example 5. Find and sketch the image of the rectangle with vertices (0, 0), (1, 0), (1, l), and (0, 1) for h = 2 and k = 3.
Refer to the discussion following Example 3 to develop the double angle identities for sine and cosine by using the matrix transformation f(f(u)) = A(Au), where
Use a procedure similar to the one discussed after Example 3 to develop sine and cosine expressions for the difference of two angles; θ1 - θ2.
Let R be the rectangle with vertices (1, 1), (1. 4), (3, 1), and (3, 4). Let f be the shear in the x-direction with k = 3. Find and sketch the image of R.
The matrix transformation f: R2 †’ R2 defined by f(v) = Av, whereand A: is a real number, is called dilation if k > 1 and contraction if 0 (a) k = 4; (b) k = 1/4.
The matrix transformation f: R2 †’ R2 defined by f(v) = Av, whereand A: is a real number, is called dilation in the y-direction if k > 1 and contraction in the y-direction if 0

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