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Applied Linear Algebra 1st edition Peter J. Olver, Cheri Shakiban - Solutions

Let V be an inner product space and v ∈ V a fixed element. Prove that the set of all vectors w ∈ V that are orthogonal to v forms a subspace of V.
(a) Show that the polynomials p1(x) = 1, p2(x) - 1/2 - p3(x) = x2 - x + 1/6 are mutually orthogonal with respect to the L2 inner product on the interval [0, 1 ]. (b) Show that the functions sin n π x, n = 1, 2, 3, .... are mutually orthogonal with respect to the same inner product.
Determine all pairs among the functions 1,.x. cos π x, sin π x,ex, that are orthogonal with respect to the L2 inner product on [ - 1, 1 ].
Prove that the points (0,0.0), (1, 1.0), (1,0, 1), (0. 1.1) form the vertices of a regular tetrahedron, meaning that all sides have the same length. What is the common Euclidean angle between the edges? What is the angle between any two rays going from the center ( 1/2,1/2,1/2) to the vertices?
Use the dot product on R3 to answer the following: (a) Find the angle between the vectors (1,2,3) and (1.-1,2). (b) Verify the Cauchy-Schwarz and triangle inequalities for these two particular vectors. (c) Find all vectors that are orthogonal to both of these vectors.
Verify the triangle inequality for each pair of vectors in Exercise 3.2.1.
Verify the triangle inequality for the vectors and inner products in Exercise 3.2.4.
Verify the triangle inequality for the functions in Exercise 3.2.12 for the indicated inner products.
Verify the triangle inequality for the two particular functions appearing in Exercise 3.1.30 with respect to the L2 inner product on (a) The unit square (b) The unit disk
Use the L2 inner productto answer the following:(a) Find the "angle" between the functions 1 and x. Are they orthogonal?(b) Verify the Cauchy-Schwarz and triangle inequalities for these two functions.(c) Find all quadratic polynomials p(x) = Î± + bx + cx2 that are orthogonal to both of these
(a) Write down the explicit formulae for the Cauchy-Schwarz and triangle inequalities based on the weighted inner product.(b) Verify that the inequalities hold when fix) = 1, g(x) = ex by direct computation. (c) What is the "angle" between these two functions in this inner product?
Answer Exercise 3.2.37 for the Sobolev H1 inner product
Prove that ||v - w|| ≥ | ||v|| - ||w|| |. Interpret this result pictorially.
Verify the Cauchy-Schwarz inequality for the vectors v = (1. 2)T, w = (1, -3)T, using(a) The dot product(b) The weighted inner product(v,w) = vi w1 + 2v2w2(c) The inner product (3.9)
True or false: ||w|| ≤ ||v|| + ||v + w|| for any v. w ∈ V.
(a) Prove that the space Rˆž consisting of all infinite sequences x = (x1,x2,x3,...) of real numbers Xi ˆˆ R forms a vector space.(b) Prove that the set of all sequences x such thatForm a subspace. Commonly de- noted „“2 Š‚ Rˆž.(c) Write down two examples of sequences x belonging to
Verify the Cauchy-Schwarz inequality for the vectors v = (3, - 1. 2)T. w = (1, - 1. l)T. using(a) The dot product(b) The weighted inner product(v. w) = v1w1 + 2v2w2 + 3 v3w3(c) The inner product
Use the Cauchy-Schwarz inequality to prove (a cos θ + b sin θ)2 ≤ a2 + b2 for any 9,a,b.
Prove that (ai + a2 +....... + an)2 ≤ n(a21 + a22 + .......+ a2n) for any real numbers a\..........a". When does equality hold?
The Law of Cosines: Prove that the formula ||v - w||2 = ||v||2 + ||w||2 - 2 ||v|| ||w|| cosθ, (3.21) where θ is the angle between v and w, is valid in any inner product space.
Explain why the inequality (v, w) < ||v|| ||w||, obtained by omitting the absolute value sign on the left hand side of Cauchy-Schwarz, is valid.
Compute the 1, 2, 3 and oo norms of the vectorsVerify the triangle inequality in each case.
Prove that the following formulas define norms on R2:(a) ||v|| = √221 + 3v22(b) ||v|| = √2v21] - v1v2 + 2v22(c) ||v|| =2|v1| + |v2|(d) ||v|| = max{ 2|v1|, |v2| }(e) ||v|| = max{ |v1, - v2|, |v1 + v2|}(f) ||v|| = |v1, - v2| + v1, + v2|
Which of the following formulas define norms on R3? (a) ||v|| = √2v21 + v22 + 3v23 (b) ||v|| = √v21 + 2v1v2 + v22 + v23 (c) ||v|| = max {v1,|, |v2|, |v3|} (d) ||v|| = {v1, - v2| + |v2 - v3| + |v3 - v1} (e) ||v|| = |v1 - v2| + { |v1|, |v2|}
Prove that two parallel vectors v and w have the same norm if and only if v = ± w.
Prove that the ∞ norm on R2 does not come from an inner product. Hint: Look at Exercise 3.1.12.
Can formula (3.11) be used to define an inner product for (a) The 1 norm ||v||1 on R2? (b) The ∞ norm ||v||∞ on R2?
Prove that lim ||v||n = ||v||∞ for any v ∈ R2.
Justify the triangle inequality for (a) The L1 norm (3.31); (b) The L∞ norm (3.32)
Let w(x) > 0 for a (a) Prove thatdefines a norm on C°[«. h ]. called the weighted L1 norm. (b) Do the same for the weighted Lˆž norm ||f||ˆž.u, = max{|f(x)| w(x) ; a ‰¤ x ‰¤ h }.
Let || • ||1 and || • ||2 be two different norms on a vector space V.(a) Prove that ||v|| = max{ ||v||1. ||v||2} defines a norm on V.(b) Does ||v|| = min{ ||v||1. ||v||2} define a norm?(c) Does the arithmetic mean||v|| = 1/2(||v||1 + ||v||2)define a norm?(d) Does the geometric mean||v|| =
Answer Exercise 3.3.1 fora.b. c.
Find a unit vector in the same direction as v = (1.2, -3)T for (a) The Euclidean norm (b) The weighted norm ||v||2 = 2v1/2 + v22 + 1/3 u23 (c) The I norm (d) The ∞ norm (e) The norm based on the inner product 2v1w1 - vl)| w2 - v2w1) + 2v2w2 - v2w3- v3w2 + 2v1wt,
Show for any choice of given angles 0, f and f, the following are unit vectors in the Euclidean norm:(a) (cos θ cosθ,cosθ sin∅,sinθ)T(b) 1/√2 (cosθ,sinθ,cos∅,sinθ)T(c) (cos θ cos ∅ cos
Plot the unit circle (sphere) for (a) The weighted norm ||v|| = √v21 + 4 v22; (b) The norm based on the inner product (3.9); (c) The norm of Exercise 3.3.9.
Draw the unit circle for each norm in Exercise 3.3.10.
Sketch the unit sphere S| C R3 for (a) The L1 norm. (b) The norm, (c) The weighted norm ||v||2 = 2v21 + v22 + 3v23, (d) The norm ||v|| = max{ |v1 + v2, |v1 + v3|. |v2 + v3| ).
True or false: Two norms on a vector space have the same unit sphere if and only if they are the same norm.
Find the unit function that is a constant multiple of the function f(x) = .v - 4 with respect to the (a) L1 norm on 10, 1 ] (b) L2 norm on [ 0. 1 ] (c) L∞ norm on [0. 1 ] (d) L1 norm on [ - 1. 1 ] (e) L2 norm on [ - 1. 1 ] (f) L∞ norm on [ - 1. 1 ]
Which two of the vectors u = (-2,2, 1)T, v = (1,4, l)T, w = (0,0, -l)r are closest to each other in distance for (a) The Euclidean norm? (b) The ∞ norm? (c) The 1 norm?
A subset S C R" is called convex if, for any x, y ∈ S, the line segment joining x to y is also in S, i.e., t x + (1 - t) y ∈ S for all 0 ≤ t ≤ 1. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?
Check the validity of the inequalities (3.38) for the particular vectors (a) (1,-1)T (b) (1. 2, 3)T (c) (1,1, l,l)T (d) (1,-1, -2,-1, l)T
Find all v ∈ R2 such that (a) ||v||1 = ||v||∞ (b) ||v||1 = ||v||∞ (c) ||v||2 = ||v||∞ (d) ||v||∞ = 1/√2 ||v||
How would you quantify the following statement: The norm of a vector is small if and only if all its entries are small.
Can you find an elementary proof of the inequalities ||v||∞ ≤ || v||2 ≤ √n ||v||∞ for v ∈ R" directly from the formulas for the norms?
(i) Show the equivalence of the Euclidean norm and the 1 norm on Rn by proving ||v||2 ≤ ||v|| ≤ √n ||v||2.(ii) Verify that the vectors in Exercise 3.3.31 satisfy both inequalities.(iii) For which vectors v € R" is(a) ||v||2 = ||v||1?(b) ||v||h = √n ||v||2?
(a) Establish the equivalence inequalities (3.35) between the 1 and oo norms.(b) Verify them for the vectors in Exercise 3.3.31.(c) For which vectors v ∈ R" are your inequalities equality?
Let ||.||2 denote the usual Euclidean norm on Rn. Determine the constants in the norm equivalence inequalities c* ||v|| ≤ ||v||2 ≤ C* ||v|| for the following norms: (a) The weighted norm ||v|| = √2v21 + 3v22, (b) The norm ||v|| = max{ |v1+ v22|, |v1, - v2| }.
What does it mean if the constants defined in (3.36) are equal: c' = C*?
(a) Compute the L∞ norm on [0, 1 ] of the functions f(x) = 1/3 - x and g(x) = x - x2. (b) Verify the triangle inequality for these two particular functions.
Prove that if [a, b] is a bounded interval and f € C°[a. b], then ||f||2 ≤ √b - a ||f||∞.
In this exercise, the indicated function norms are taken over all of R.a.||fn|| = 1. but ||fn||2 †’ ˆž as n †’ ˆž(b) Explain why there is no constant C such that ||f||2 ‰¤ C || f ||ˆž for all functions f.(c) that || f || = 1. but ||fn||ˆž as n †’ 00. Conclude that there is
(a) Prove that the L∞ and L2 norms on the vector space C°[- 1, 1 ] are not equivalent. Hint: Look at Exercise 3.3.41 for ideas.(b) Can you establish a bound in either direction, i-e.. ||f||∞ ≤ C ||f||2 or ||f||2 ≤ C ||f||∞ for all ∈ C°[- 1, 1 ] for some positive constant C, C?
Suppose (v. w), and (v, w)2 are two inner products on the same vector space V. For which or. α β ∈ R is the linear combination (v. w) = a(v, w), + β (v, w)2 a legitimate inner product? Hint: The case when a, β ≥ 0 is easy. However, some negative values are also permitted, and your task is
Answer Exercise 3.3.4 using the L1 norm
Which two of the functions f(x) = 1, g(x) = x, h(a) = sin π x are closest to each other on the interval [0, 1 ] under(a) The L1 norm? (b) The L2 norm? (c) The Lx norm?
Consider the functions f(x) = 1 and g(x) = x - 3/4 as elements of the vector space C°[0, 1]. For each of the indicated norms, compute ||f||, ||g||, ||f + g||. and verify the triangle inequality: (a) The L1 norm (b) The L2 norm (c) The L∞ norm (d) The L∞ norm
Answer Exercise 3.3.7 when f(x) = ex and g(x) = e~x.
Carefully prove that ||(a, y)T|| = |a| + 2 |x - y| defines a norm on R2.
Which of the following 2 x 2 matrices are positive definite?(a)(b) (c) (d) (e) (f) In the positive definite cases, write down the formula for the associated inner product.
Let A" be a nonsingular symmetric matrix. (a) Show that xTK~1x = yTK y, where Ky = x. (b) Prove that if K is positive definite, so is K-1.
Prove that an n x n symmetric matrix K is positive definite if and only if, for every O ≠ v ∈ R", the vectors v and K v meet at an acute Euclidean angle: |θ| < 1/2π.
Prove that the inner product associated with a positive definite quadratic form q(x) is given by the polarization formula (x. y) = 1/2[q(x + y) - q(x) - q(y) ].
(a) Is it possible for a quadratic form to be positive, q(x+) > 0, at only one point x + ∈ Rn?(b) Under what conditions is q(x0) = 0 at only one point?
(a) Show that a symmetric matrix N is negative definite if and only if K = - N is positive definite.(b) Write down two explicit criteria that tell whether or not a 2 x 2 matrixis negative definite. (c) Use your criteria to check whether (i) (ii) (iii) are negative definite.
Show that x = (11) is a null direation forbut x ˆˆ ker K.
Explain why an indefinite quadratic form necessarily has a non-zero null direction.
Let K = KT. True or false: (a) If K admits a null direction, then ker K ≠ {0}. (b) If K has no null directions, then K is either positive or negative definite.
In special relativity, light rays in Minkowski space- time M'1 travel along the light cone which, by definition, consists of all null directions associated with an indefinite quadratic form q(x) = xT Kx. Find and sketch a picture of the light cone when the coefficient matrix K isa.b. c. Remark: In
(a) Let K and L be symmetric n x n matrices. Prove that xTKx = xTx for all x ∈ Rn if and only if K = L.(b) Find an example of two non-sy mmetric matrices K ≠ L such that xTKx = xTLx for allX € Rn.
Supposeis a general quadratic form on ?. . whose coefficient matrix A is not necessarily symmetric. Prove that r/(x) = xTKx. where K = 1/2 (A + AT) lisa symmetric matrix. Therefore, we do not lose any generality by restricting our discussion to quadratic forms that are constructed from symmetric
A function f(x) on Rn is called homogeneous of degree k if f(cx) = ci(x) for all scalars c.(a) If a ˆˆ Rn is a fixed vector, show that a linear form £(x) = a . x = a1X1 +-----+ anxn is homogeneous of degree 1.(b) Show that a quadratic formis homogeneous of degree 2.(c) Find a homogeneous
(a) Find the Gram matrix corresponding to each of the following sets of vectors using the Euclidean dot product on Rn.(b) Which are positive definite? (c) If the matrix is positive semi-definite, find all its null vectors
Recompute the Gram matrices for cases (iii)-(v) in the previous exercise using the weighted inner product (x, y) = x1y1 +2x2y2 + 3x3y3- Does this change its positive definiteness?
Recompute the Gram matrices for cases (vi)-(viii) in Exercise 3.4.22 for the weighted inner product (x,y) = x1,y1, + 1/2x2y2 + 1/3X3y3 + 1/4 x4y4.
Find the Gram matrix K for the functions l,ex, e2x using the L2 inner product on [0, 1 ]. Is K- positive definite?
Answer Exercise 3.4.25 using the weighted inner product (/, g) =
Find the Gram matrix K for the monomials 1, x, x2, x3 using the L2 inner product on [ - 1, 1 ]. Is K positive definite?
Answer Exercise 3.4.27 using the weighted inner product (f,g)
Prove that every positive definite matrix K can be written as a Gram matrix.
(a) Prove that a diagonal matrix D = diag(c1,c2, ...,cn) is positive definite if and only if all its diagonal entries are positive: c, > 0.(b) Write down and identify the associated inner product.
(a) Prove that if K is a positive definite matrix, then K2 is also positive definite.(b) More generally, prove that if S = ST is symmetric and nonsingular, then S2 is positive definite.
Let K - ATCA, where C > 0. Prove that(a) ker K = coker K = kerA;(b) mg K -- comg K = comg A
Show that z is a null vector for the quadratic form q (x) = xT Kx based on the Gram matrix K = ATA if and only if z ∈ ker K.
Let A be an m x n matrix.(a) Explain why the product L = AAT is a Gram matrix.(b) Show that, even though they may be of different sizes, both Gram matrices K = ATA and L = AAT have the same rank.(c) Under what conditions are both K and L positive definite?
Suppose K is the Gram matrix computed from V1,. " , v" ∈ V relative to a given inner product. Let K be the Gram matrix for the same elements, but computed relative to a different inner product. Show that K > 0 if and only if -K >0.
Let K1| = AT1C1A1 and K2 = AT2A2 be any two n x n Gram matrices. Let K = K1+ K2.(a) Show that if K1 + K2 > 0 then K > 0.(b) Give an example where K1 and K2 are not positive definite, but K > 0.(c) Show that K = ATCA is also a Gram matrix. Hint: : A will have size (m1 + m2) x n, where mi
(a) Show that every diagonal entry of a positive definite matrix must be positive. (b) Write down a symmetric matrix with all positive diagonal entries that is not positive definite. (c) Find a nonzero matrix with one or more zero diagonal entries that is positive semi-definite.
Write out the Cauchy-Schwarz and triangle inequalities for the inner product defined in Example 3.22.
Prove that if K is any positive definite matrix, then any positive scalar multiple c K, c > 0, is also positive definite.
(a) Show that if K and L are positive definite matrices, so is K + L.(b) Give an example of two matrices that are not positive definite whose sum is positive definite.
Find a pair of positive definite matrices K and L whose product K L is not positive definite.
(a) Prove that a positive definite matrix has positive determinant: det K > 0.(b) Show that a positive definite matrix has positive trace: tr K >0.(c) Show that every 2 x 2 symmetric matrix with positive determinant and positive trace is positive definite.(d) Find a symmetric 3 x 3 matrix
(a) Prove that if K1, K2 are positive definite n x n matrices, then KIs a positive definite 2n Ã— 2 n matrix. (b) is the converse true?
Let ||∙|| be any norm on Rn.(a) Show that q(x) is a positive definite quadratic form if and only if q(u) > 0 for all unit vectors, ||u|| = 1.(b) Prove that if S = ST is any symmetric matrix, then K = S + c I > 0 is positive definite for c»0 sufficiently large.
Prove that every symmetric matrix S = K + N can be written as the sum of a positive definite matrix K and a negative definite matrix N. Hint: Use Exercise 3.5.12(b)
(a) Prove that any regular symmetric matrix can be decomposed as a linear combinationK = d1,11IT1 + d2I21T2 + €¢ €¢ €¢ + dn11lT1 (3.69)of symmetric rank 1 matrices, as Exercise 1.8.15. where 11.......1n are the columns of the special lower triangular matrix L and
There is an alternative criterion for positive definiteness based on subdeterminants of the matrix. The 2x2 version already appears in (3.62).(a) Prove that a 3 x 3 matrixIs positive definite if and only if Î± > 0, ad - b2 > 0 det K > 0. (b) Prove the general version: an n x n matrix K
Let K be a symmetric matrix. Prove ha if a non- positive diagonal entry appears anywhere (not nece ssarily in the pivot position) in the matrix during Regular Gaussian Elimination. (hen K is not posiL ive definite.
Formulate a determinantal criterion similar to Exercise 3.5.15 for negative definite marices. Write out the 2 x 2 and 3 x 3 cases explicitly.

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