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mathematics
applied linear algebra

Applied Linear Algebra 2nd Edition Peter J. Olver, Chehrzad Shakiban - Solutions

True or false: If ΙΙ v + w ΙΙ = ΙΙ v ΙΙ + ΙΙ w ΙΙ, then v, w are parallel vectors.
(a) Find a such that (2, a,−3)T is orthogonal to (−1, 3,−2)T. (b) Is there any value of a for which (2, a,−3 )T is parallel to (−1, 3,−2)T?
Find all vectors in R3 that are orthogonal to both (1, 2, 3)T and (−2, 0, 1)T.
Prove that limp → ∞ ΙΙ v ΙΙp = ΙΙ v ΙΙ∞ for all v ∈ R2.
Suppose (v ,w) defines an inner product on a vector space V. Explain why it also defines an inner product on every subspace W ⊂ V.
True or false: A negative definite matrix must have negative trace and negative determinant.
For what values of a, b are the vectors (1, 1, a)T and (b,−1, 1)T orthogonal(a) With respect to the dot product?(b) With respect to the weighted inner product of Exercise 3.2.17?Data From Exercise 3.2.17Answer Exercises 3.2.15 and 3.2.16 using the weighted inner product (v ,w) = 3ν1ω1 + 2ν2ω2
Write out the real and imaginary parts of the power function xc with complex exponent c = a + ib ∈ C.
Find a non-zero quadratic polynomial that is orthogonal to both p1(x) = 1 and p2(x) = x under the L2 inner product on the interval [−1, 1].
Let ΙΙ · ΙΙ be a norm on Rn. Prove that there is a constant C > 0 such that the entries of every v = (ν1, ν2, . . . , νn)T ∈ Rn are all bounded, in absolute value, by ΙΙ νi ΙΙ ≤ C ΙΙ v ΙΙ.
Prove that ΙΙ v − w ΙΙ ≥ ΙΙ v ΙΙ − ΙΙ w ΙΙ. Interpret this result pictorially.
Write down all diagonal n × n orthogonal matrices.
Let K be a 2 × 2 Gram matrix. Explain why the positive definiteness criterion (3.55) is equivalent to the Cauchy–Schwarz inequality. a > 0, ac-b²0,
Find all quadratic polynomials that are orthogonal to the function ex with respect to the L2 inner product on the interval [0, 1].
How many unit vectors are parallel to a given vector v ≠ 0? (a) 1,(b) 2,(c) 3,(d) ∞,(e) Depends on the norm.
Find all v ∈ R2 such that(a) ΙΙ v ΙΙ1 = ΙΙ v ΙΙ∞,(b) ΙΙ v ΙΙ1 = ΙΙ v ΙΙ2,(c) ΙΙ v ΙΙ2 = ΙΙ v ΙΙ∞,(d) ΙΙ v ΙΙ∞ = 1 / √2 ΙΙ v ΙΙ2.
True or false: The set of all complex-valued functions u(x) = ν(x) + iw(x) with u(0) = i is a subspace of the vector space of complex valued functions.
Suppose ΙΙ · ΙΙ1, ΙΙ · ΙΙ2 are two norms on Rn. Prove that the corresponding matrix norms satisfy for any n × n matrix A for some positive constants c* ||A||1 < ||A|2
True or false: If B = S−1AS are similar matrices, then ΙΙ B ΙΙ∞ = ΙΙ A ΙΙ∞.
Find a matrix A such that ΙΙ A2 ΙΙ∞ ≠ ΙΙ A ΙΙ2∞.
Let V be a complex normed vector space. How many unit vectors are parallel to a given vector 0 ≠ v ∈ V? (a) None;(b) 1; (c) 2; (d) 3; (e) ∞; (f) Depends upon the vector;(g) Depends on the norm.
(a) Prove that, for all θ,ϕ,ψ,is a proper orthogonal matrix.(b) Write down a formula for Q−1. Remark. It can be shown that every proper orthogonal matrix can be parameterized in this manner; θ,ϕ,ψ are known as the Euler angles, and play an important role in applications in mechanics and
Prove that if p(t) = p(−t) is an even polynomial, then all the odd-order coefficients c2j+1 = 0 in its Legendre expansion (4.56) vanish. p(t) = co do(t)+c₁ q₁ (t) + + Cn 9n (t).
(a) How many orthonormal bases does R have? (b) What about R2? (c) Does your answer change if you use a different inner product? Justify your answers.
Find the orthogonal projection of v = (1, 2, −1, 2)T onto the span of (1,−1, 2, 5)T and (2, 1, 0,−1)T using the weighted inner product (v , w) = 4ν1ω1 + 3ν2ω2 + 2ν3ω3 + ν4ω4.
True or false: Reordering the original basis before starting the Gram–Schmidt process leads to the same orthogonal basis.
Prove the formula for in (4.73) . ||PK||
True or false: The standard algorithm for finding a basis for kerA will always produce an orthogonal basis.
Show that if A = AT is a symmetric matrix, then Ax = b has a solution if and only if b is orthogonal to ker A.
Let u1, . . . , un and û1, . . . , ûn be orthonormal bases of an inner product space V.Prove that is an orthogonal matrix. ū₂ = n Σ qij¹j j=1 qiju, for i = 1,..., n, where Q = (ij)
True or false: Applying an elementary row operation to an orthogonal matrix produces an orthogonal matrix.
Let W ⊂ V. Prove that(a) W ∩W⊥ = {0},(b) W ⊆ (W⊥)⊥.
(a) Write down the Householder matrices corresponding to the following unit vectors:(b) Find all vectors fixed by a Householder matrix, i.e., Hv = v — first for the matrices in part (a), and then in general. (c) Is a Householder matrix a proper or improper orthogonal matrix? T 34 (i) (1,0), (ii)
Let Hn = QnRn be the QR factorization of the n × n Hilbert matrix (1.72). (a) Find Qn and Rn for n = 2, 3, 4. (b) Use a computer to find Qn and Rn for n = 10 and 20.(c) Let x⋆ ∈ Rn denote the vector whose ith entry is x⋆i = (−1)i i/(i + 1). For the values of n in parts (a) and (b),
For the potential energy function in (5.1), where is the equilibrium position of the ball? f(x, y) = 3x² - 2xy + 4y² + x − 2y + 1,
Find the closest point in the plane spanned by (1, 2,−1)T, (0,−1, 3)T to the point (1, 1, 1)T. What is the distance between the point and the plane?
The median price (in thousands of dollars) of existing homes in a certain metropolitan area from 1989 to 1999 was:(a) Find an equation of the least squares line for these data. (b) Estimate the median price of a house in the year 2005, and the year 2010, assuming that the trend continues. year
Find the minimizer of the function f(x, y) = (3x − 2y + 1)2 + (2x + y + 2)2.
Find the closest point or points to b = (−1, 2)T that lie on(a) The x-axis,(b) The y-axis,(c) The line y = x,(d) The line x + y = 0,(e) The line 2x + y = 0.
Solve Exercise 5.1.3 when distance is measured in (i) the ∞ norm, (ii) the 1 norm.Data From Exercise 5.1.3Find the closest point or points to b = (−1, 2)T that lie on(a) The x-axis,(b) The y-axis,(c) The line y = x,(d) The line x + y = 0,(e) The line 2x + y = 0.
For the data points in Exercise 5.5.9, determine the plane z = α + βx + γy that fits the data in the least squares sense when the errors are weighted according to the reciprocal of the distance of the point (xi, yi, zi) from the origin.Data From Exercise 5.5.9For the data points(a) Determine the
(a) Generalize Exercise 5.1.8 to find the distance between a point (x0, y0, z0)T and the plane ax + by + cz + d = 0 in R3. (b) Use your formula to compute the distance between (1, 1, 1)T and the plane 3x − 2y + z = 1.Data From Exercise 5.1.8(a) Prove that the distance from the point (x0, y0)T
Let ql(x) denote the trigonometric polynomial (5.119) obtained by summing the first 2l + 1 discrete Fourier modes. Suppose the criterion for compression of a signal f(x) is that For the particular function in Exercise 5.6.10, how large do you need to choose k when ε = .1? ε = .01? ε = .001?Data
(a) Prove that the distance from the point (x0, y0)T to the line ax + by = 0 is .(b) What is the minimum distance to the line ax + by + c = 0? Taxo+byol √a² +6²
Find the vector w⋆ ∈ span (0, 0, 1, 1), (2, 1, 1, 1) that minimizes ΙΙw − (0, 3, 1, 2)ΙΙ.
(a) Find the distance from the point b = (1, 2,−1)T to the plane x − 2y + z = 0.(b) Find the distance to the plane x − 2y + z = 3.
Find the closest point to the vector b = (1, 0, 2)T belonging the two-dimensional subspace spanned by the orthogonal vectors v1 = (1,−1, 1 )T, v2 = (−1, 1, 2 )T.
The following table gives the population of the United States for the years 1900-2000.(a) Use an exponential growth model of the form y = ce at to predict the population in 2020, 2050, and 3000. (b) The actual population for the year 2020 has recently been estimated to be 334 million. How does
True or false: The minimal-norm solution to Ax = b is obtained by setting all the free variables to zero.
Let b = (0, 3, 1, 2)T. Find the vector w⋆ ∈ span {(0, 0, 1, 1)T, (2, 1, 1,−1 )T} such that ΙΙ w* − b ΙΙ is minimized.
The table measures the altitude of a falling parachutist before her chute has opened. Predict how many seconds she can wait before reaching the minimum altitude of 1500 meters. time in sec meters 0 4500 10 4300 20 3930 30 3000
A missile is launched in your direction. Using a range finder, you measure its altitude at the times: How long until you have to run? time in sec altitude in meters 0 200 10 650 20 30 40 970 1200 1375 50 1130
Justify the fact that the orthogonal sample vector qk in (5.62) is a linear combination of only the first k monomial sample vectors. ak = (ak (t₁), ..., 9 k. (tm)) = Cko to + Ck₁t₁+· 9k +ekktk
Given A as in (5.50) with m A = 1 t₁ t 1 t2 m ... t₁ th : th 'm X = α0 α1 α2 an n y = Y1 Y2 Ym
Re-solve Exercise 5.5.15 using the respective weights 2, 1, .5 at the three data points.Data From Exercise 5.5.15Given (a) Find the straight line y = α + βt that best fits the data in the least squares sense;(b) find the parabola y = α + βt + γt2 that best fits the data. Interpret the error.
An individual bar in a structure experiences a stress of 3 under a unit horizontal force applied to all the nodes and a stress of −2 under a unit vertical force applied to all nodes. What combinations of horizontal and vertical forces will make the bar stress-free?
Let f(x) = x. Find the trigonometric function of the form g(x) = a + b cos x + c sin x that minimizes the L2 error π ||g-f || = =√√ [9(r) - f(x)]² dr.
Find the hyperbolic function g(t) = a cosh t + b sinh t that best approximates the data in Exercise 5.5.41.Data From 5.5.41Given the values construct the trigonometric function of the form g(t) = a cos πt + b sin πt that best approximates the data in the least squares sense. ti Yi 0 1 5 .5 1 .25
If a bar in a structure compresses 2 cm under a force of 5 newtons applied to a node, how far will it compress under a force of 20 newtons applied at the same node?
Find the radial polynomial p(x, y) = a + br + cr2, where r2 = x2 + y2, that best approximates the function f(x, y) = x using the L2 norm on the unit disk D = {r ≤ 1} to measure the least squares error.
Consider an electrical network running along the sides of a tetrahedron. Suppose that each wire contains a 3 ohm resistor and there is a 10 volt battery source on one wire. Determine how much current flows through the wire directly opposite the battery.
Suppose all bars have unit stiffness. Explain why the internal forces in a structure form the solution of minimal Euclidean norm among all solutions to AT y = f.
Suppose that when subject to a nonzero external force f ≠ 0, a mass–spring chain has equilibrium position u⋆. Prove that the potential energy is strictly negative at equilibrium: p(u⋆) < 0.
What is the analogue of condition (6.33) for a disconnected graph? z. f = f₁ + f₂ + ... + fn = 0,
True or false: If a structure constructed out of bars with identical stiff nesses is stable, then the same structure constructed out of bars with differing stiff nesses is also stable.
Determine which of the following sets of vectors are bases of R2:(a)(b)(c)(d)(e) (13), -3 -2 5
Which of the following sets of vectors span all of R2?(a) (b)(c)(d)(e)(f) 1 -1
Determine which of the following are bases of R3:(a)(b)(c)(d) 1 (-3), (¯ -2 5 ст
List the diagonal entries of A = 97 2.6 13 37 48 5 9 10 11 14 15 8 12 16,
Is there a matrix analogue of formula (1.56), namely ATB = BTA? vw = (vw) = w²v, W
Let Compute AT and BT. Then compute (AB)T and (BA)T without first computing AB or BA. 3 -1 A ² 1 = (₁ - 1² - 1), B = 2 -1 -1 21 ܟܕ ܗܘ 20 -3 4
Find all matrices B that commute (under matrix multiplication) with = (1 ²). A =
Let Given thatandfind a solution to A = 2 2 1 2 5 1 3 -1 -1 2
Determine all values of the scalar k for which the following four matrices form a basis for M2 × 2: 4₁ = (₁ (1 0 k -3 1 0 0 k - 0)₁ 4₂ = (₁-0), 43=(-²2): 4₁=(-2-2) -1), 1), A3 A4 1 -k -1
Which of the following are subspaces of R3? Justify your answers! (a) The set of all vectors ( x, y, z )T satisfying x + y + z + 1 = 0. (b) The set of vectors of the form ( t,−t, 0 )T for t ∈ R. (c) The set of vectors of the form ( r − s, r + 2s,−s )T for r, s
Let S = {0, 1, 2, 3}. (a) Find the sample vectors corresponding to the functions 1, cos πx, cos 2πx, cos 3πx. (b) Is a function uniquely determined by its sample values?
True or false: An interval is a vector space.
Find two different functions f(x) and g(x) that have the same sample vectors f, g at the sample points x1 = 0, x2 = 1, x3 = −1.
(a) Let x1 = 0, x2 = 1. Find the unique linear function f(x) = ax + b that has the sample vector f = (3, − 1)T. (b) Let x1 = 0, x2 = 1, x3 = −1. Find the unique quadratic function f(x) = ax2 + bx + c with sample vector f = (1,−2, 0 )T.
Determine which of the following sets of vectors x = (x1, x2, . . . , xn)T are subspaces of Rn:(a) All equal entries x1 = · · · = xn; (b) All positive entries: xi ≥ 0; (c) First and last entries equal to zero: x1 = xn = 0; (d) Entries add up to zero: x1 + · · · + xn =
Which of the following are subspaces of the vector space of n × n matrices Mn×n? The set of all (a) Regular matrices; (b) Nonsingular matrices; (c) Singular matrices;(d) Lower triangular matrices; (e) Lower unitriangular matrices; (f) Diagonal matrices;(g) Symmetric
Prove that a vector space has only one zero element 0.
Determine which of the following conditions describe subspaces of the vector space C1 consisting of all continuously differentiable scalar functions f(x).(a) f(2) = f(3),(b) f′(2) = f(3),(c) f′(x) + f(x) = 0,(d) f(2 − x) = f(x),(e) f(x + 2) = f(x) + 2,(f) f(−x) = ex f(x).(g) f(x) = a + b
A physical apparatus moves 2 meters under a force of 4 newtons. Assuming linearity, how far will it move under a force of 10 newtons?
True or false: The zero vector belongs to the span of any collection of vectors.
Applying a unit external force in the horizontal direction moves a mass 3 units to the right, while applying a unit force in the vertical direction moves it up 2 units. Assuming linearity, where will the mass move under the applied force f = (2,−3 )T ?
Suppose x⋆1 and x⋆2 are both solutions to Ax = b. List all linear combinations of x⋆1 and x⋆2 that solve the system.
Show that the set of solutions to u′′ = x + u does not form a vector space.
True or false: If x⋆1 solves Ax = c, and x⋆2 solves B x = d, then x⋆ = x⋆1 + x⋆2 solves (A + B)x = c + d.
Under what conditions on the coefficient matrix A will the systems in (2.34) all have a solution?
True or false: The six 3 × 3 permutation matrices (1.30) are linearly independent.
True or false: A set of vectors is linearly dependent if the zero vector belongs to their span.
Does a single vector ever define a linearly dependent set?
(a) Determine whether the polynomials ƒ1(x) = x2 −3, ƒ2(x) = 2 − x, ƒ3(x) = (x − 1)2, are linearly independent or linearly dependent.(b) Do they span the vector space of all quadratic polynomials?
Can you devise a nonzero matrix whose row echelon form is the same as the row echelon form of its transpose?
Prove or give a counterexample: If U is the row echelon form of A, then img U = img A.
True or false: If kerA = kerB, then rankA = rankB.
Determine the rank of the following matrices:(a)(b)(c)(d)(e)(f)(g)(h)(i) 1 1-2
Determine whether the following matrices are singular or nonsingular:(a)(b)(c)(d)(e)(f)(g)(h) 1 12
Which of the following pairs of matrices commute under matrix multiplication? Compute the indicated combinations where possible.(a)(b)(c) 1 2 (-22), ( ²), 1 5 3 0

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