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

Show that if v1......... vn span V ≠ {0}, then one can choose a subset vil,........vim that forms a basis of V. Thus, dim V = m ≤ n. Under what conditions is dim V = n?
(a) Prove that if v1,..., vm forms a basis for V ⊆ Rn, then m < n.(b) Under the hypotheses of part (b). prove that there exist vectors vm+1. . .,vn e Rn \ V such that the complete collection v1,..., vn forms a basis for Rn.(c) Illustrate by constructing bases of R3 that include(i) the basis
Let W ⊂ V be a subspace. (a) Prove that dim W < dim V. (b) Prove that if dim W = dim V = n < ∞, then W = V. Equivalently, if W ⊆ V is a proper subspace of a finite-dimensional vector space, then dim W < dim V. (c) Give an example where the result is false if dim V = ∞.
Let W. Z ⊂ V be complementary subspaces in a vector space V, as in Exercise 2.2.24. (a) Prove that if (w1,...,wj} form a basis for W and {z1.......zk} a basis for Z, then {w1,..., wj, z1,.... zk} form a basis for V. (b) Prove that dim W + dim Z = dim V.
Let f1(x),..., fn(x) be scalar functions. Suppose that every set of sample points x1, . . . ,xm ∈ R, for all finite m ≥ 1, leads to linearly dependent sample vectors f1.......fn ∈ Rm. Prove that f1(x),..., fn(x) are linearly dependent functions.
Find a basis for (a) The plane given by the equation z -2 y = 0 in R3; (b) The plane given by the equation 4x+3y-z = 0 in R3; (c) The hyperplane x + 2y + z - u> = 0 in R4.
Let(a) Do v1. v2, v3, v4 span R3? Why or why not? (b) Are v1. v2. v3, v4 linearly independent? Why or why not? (c) Do v1, v2, v3, v4 form a basis for R3? Why or why not? If not, is it possible to choose some subset which is a basis? (d) What is the dimension of the span of v1, v2, v3, v4? Justify
Answer Exercise 2.4.4 when
(a) Show thatare two different bases for the plane x-2y-4z = 0. (b) Show how to write both elements of the second basis as linear combinations of the first. (c) Can you find a third basis?
A basis v1,...,vn of Rn is called right handed if the n Ã— n matrix A = (vi €¢€¢€¢ vn) whose columns are the basis vectors has positive determinant: det A > 0. If det A (a) Which of the following form right handed bases of R3?(i)(ii)(iii)(iv)(b) Show that if v1, v2, v3 is a left
Find a basis for and the dimension of the following subspaces:(a) The space of solutions to the linear system Ax = 0, where(b) The set of all quadratic polynomials p(x) = ax2 + bx + c that satisfy p(l) = 0. (c) The space of all solutions to the homogeneous ordinary differential equation u"' - u'' +
Find a basis for and the dimension of the span of(a)(b) (c)
Characterize the range and kernel of the following matrices:(a)(b) (c) (d)
Let A be an m x n matrix. Suppose thatis a (m + k) x n matrix whose first m rows are the same as A. Prove that kerC Š† ker A. Thus, appending more rows cannot increase the size of a matrix's kernel. Give an example where kerC ‰ ker A
Let A be an m × n matrix. Suppose that C = (A B ) is an m × (n + k) matrix whose first n columns are the same as A. Prove that mgC ⊇ mg A. Thus, appending more columns cannot decrease the size of a matrix's range. Give an example where mgC ≠ rng A.
Find the solution x*1 to the systemand the solution x*2 to Express the solution to as a linear combination of x*1 and x*2.
LetGiven that and find a solution to Ax = 2b1 + b2
(a) Show thatare particular solutions to the system (b) Find the general solution.
For each of the following matrices, write the kernel as the span of a finite number of vectors. Is the kernel a point, line, plane or all of R3?(a) (2 -I 5)(b)(c) (d) (e) (f)
Let A be a nonsingular m x m matrix.(a) Explain in detail why the solutions x*1,..., x*m to the systems (2.32) are the columns of the matrix inverse A-1.(b) Illustrate your argument in the case
For each of the following matrices find bases for the(i) range,(ii) corange,(iii) kernel, and(iv) cokemel.(a)(b)(c)(d)
Find a set of columns of the matrixthat form a basis for its range. Then express each column as a linear combination of the basis columns.
Find bases for the range and corange ofMake sure they have the same number of elements. Then write each row and column as a linear combination of the appropriate basis vectors.
For each of the following matrices A:(a) Determine the rank and the dimensions of the four fundamental subspaces.(b) Find bases for both the kernel and cokemel.(c) Find explicit conditions on vectors b which guarantee that the system Ax = b has a solution.(d) Write down a specific nonzero vector b
Find the dimension of and a basis for the subspace spanned by the following sets of vectors.(a)(b) (c) (d) (e)
Show that the set of all vectors of the v = (a - 3b. a +2c + 4d. b + 3c - d, c - d)T . where a, b.c.d are real numbers, forms a subspace of R4, and find its dimension.
Find a basis of the solution space of the following homogeneous linear systems. (a) x1 - 2x3 - 0, x2 +x4 = 0. (b) 2x1 + x2 - 3x3 + x4 = 0, 2x1 - x2 - x3 - x4 = 0. (c) x1 -x2 - 2x3 + 4x4 = 0, 2x1 +x2 - x4 = 0, - 2x1 + 2x3 - 2x4 = 0.
Find bases for the range ofusing both of the indicated methods. Demonstrate that they are indeed both bases for the same subspace by showing how to write each basis in terms of the other.
Show thatand are two bases for the same three-dimensional subspace V c R4.
(a) Find the kernel and range of the coefficient matrix for the system x -3y + 2z = a, 2x - 6y + 2w = b, z - 3 w = c. (b) Write down compatibility conditions on a, b, c for a solution to exist.
(a) Prove that if A is a symmetric matrix, then ker A = coker A and mg A = comg A.(b) Use this observation to produce bases for the four fundamental subspaces associated with(c) Is the converse to part (a) true?
Let A be a 4 × 4 matrix and let U be its row echelon form. (a) Suppose columns 1, 2. 4 of U form a basis for its range. Do columns 1, 2,4 of A form a basis for its range? If so. explain why; if not, construct a counterexample. (b) Suppose rows 1, 2, 3 of U form a basis for its corange. Do rows 1,
(a) Write down a matrix of rank r whose first r rows do not form a basis for its row space. (b) Can you find an example that can be reduced to row echelon form without any row interchanges?
Explain why the elementary row operations of types #2 and #3 do not change the corange of a matrix.
(a) Devise an alternative method for finding a basis of the corange of a matrix.(b) Use your method to find a basis for the corange ofIs it the same basis as found by the method in the text?
Prove that rng A ⊇ rng A2. More generally, prove rng A ⊇ rng(A B) for any (compatible) matrix B.
Supposeis a particular solution to the equation (a) What is b? (b) Find the general solution
Suppose A is an m x n matrix, and B and C are nonsingular matrices of sizes m × m and n × n, respectively. Prove that rank A = rank B A = rank A C = rank B AC.
Suppose A is a nonsingular n x n matrix.(a) Prove that any n x (n + k) matrix of the form (A B), where B has size n Ã— k. has rank n.(b) Prove that any (n + k) Ã— n matrix of the formwhere C has size k Ã— n, has rank n.
Let A be an m x n matrix of rank r. Suppose v1,. . . vn are a basis for Rn such that vr+1.......vn form a basis for ker A. Prove that w1 = Av1.... . wr = Avr, form a basis for mg A.
(a) Suppose A.B are m x n matrices such that ker A = ker B. Prove that there is a nonsingular m × m matrix M such that MA = B. (b) Use this to conclude that if Ax = b and Bx = c have the same solutions then they are equivalent linear systems, i.e., one can be obtained from the other by a sequence
Let A be an in x n matrix and let V be a subspace of R". (a) Show that W = AV = { A v | v € V } forms a subspace of rng A. (b) If dim V = k, show that dim W ≤ min{k, r}, where r = rank A.
(a) Show that an m × n matrix has a left inverse if and only if it has rank n. (b) Show that it has a right inverse if and only if it has rank m. (c) Conclude that only nonsingular square matrices have both left and right inverses.
Write the general solution to the following linear systems in the form (2.26). Clearly identify the particular solution x* and the element z of the kernel.(a) a - y + 3z = 1(b)(c) (d) (e) (f) (g)
Prove that the average of all the entries in each row of A is 0 if and only if (1, 1.......1 )T ∈ ker A.
Given a, r ‰ 0, characterize the kernel and the range of the matrix
A projection matrix is a square matrix P that satisfies P2 = P. (a) Prove that w ∈ mg P if and only if P w = w. (b) Show that mg P and ker P are complementary subspaces, as defined in Exercise 2.2.24, so every v ∈ Rn can be uniquely written as v = w + z where w ∈ mg P, z ∈ ker P.
True or false: If A is a square matrix, ker A ∩ mg A = (0).
Draw the digraph represented by the following incidence matrices(a)(b) (c) (d) (e)
The complete bipartite digraph Km,n is based on two disjoint sets of, respectively, m and n vertices. Each vertex in the first set is connected to each vertex in the second set by a single edge. (a) Draw K2,3, K24, and K3,3. (b) Write the incidence matrix of each digraph. (c) How many edges does
(a) Construct the incidence matrix A for the disconnected digraph D in the figure.(b) Verify that dim ker A = 3, which is the same as the number of connected components, meaning the maximal connected subgraphs in D. (c) Can you assign an interpretation to your basis for ker A ? (d) Try proving the
Prove that a graph with n nodes and n edges must have at least one circuit.
How does altering the direction of the edges of a digraph affect its incidence matrix? The cokernel of its incidence matrix? Can you realize this operation by matrix multiplication?
(a) Explain why two digraphs are equivalent under relabeling of vertices and edges if and only if their incidence matrices satisfy PAQ = B. where P. Q are permutation matrices.(b) Decide which of the following incidence matrices produce the equivalent digraphs:(i)(ii) (iii) (iv) (v) (vi) (c) How
True or false: If A and B are incidence matrices of the same size and coker A = coker 5. then the corresponding digraphs are equivalent.
(a) Explain why the incidence matrix for a disconnected graph can be written in block diagonal matrix formunder an appropriate labeling of the vertices. (b) Show how to label the vertices of the digraph in Exercise 2.6.3e so that its incidence matrix is in block form.
(a) Draw the graph corresponding to the 6 × 7 incidence matrix whose nonzero (/, j) entries equal 1 if j = i and - 1 if j = i + 1, for i = 1 to 6. (b) Find a basis for its kernel and cokemel. (c) How many circuits are in the digraph?
Write out the incidence matrix of the following digraphs.(a)(b) (c) (d) (e) (f)
For each of the digraphs in Exercise 2.6.3, see if you can predict a collection of independent circuits. Verify your prediction by constructing a suitable basis of the cokernel of the incidence matrix and identifying each basis vector with a circuit.Exercise 2.6.3(a)(b) (c) (d) (e) (f)
(a) Write down the incidence matrix A for the indicated digraph.(b) What is the rank of .4? (c) Determine the dimensions of its four fundamental subspaces. (d) Find a basis for its kernel and cokemel. (e) Determine explicit conditions on vectors b which guarantee that the system Ax = b has a
(a) Write out the incidence matrix for the cubical digraph and identify the basis of its cokernel with the circuits. (b) Find three circuits which do not correspond to any of your basis elements, and express them as a linear combination of the basis circuit vectors.
Write out the incidence matrix for the other Platonic solids: (a) Tetrahedron (b) Octahedron (c) Dodecahedron (d) Icosahedron (You will need to choose an orientation for the edges.) Show that, in each case, the number of independent circuits equals the number of faces minus i.
A connected graph is called a tree if it has no circuits.(a) Find the incidence matrix for each of the following directed trees:(i)(ii) (iii) (iv) (b) Draw all distinct trees with 4 vertices. Assign a direction to the edges, and write down the corresponding incidence matrices. (c) Prove that a
A complete graph Kn on n vertices has one edge joining every distinct pair of vertices. (a) Draw K3, K4 and K5. (b) Choose an orientation for each edge and write out the resulting incidence matrix of each digraph. (c) How many edges does Kn have? (d) How many independent circuits?
Prove that the formula (v. w) = V1W1 - V1W2 - V2W1 + bv2w2 defines an inner product on R2 if and only if b > 1.
Let V be an inner product space.(a) Prove that (x. v) = 0 for all v € V if and only if x = 0.(b) Prove that (x. v) = (y. v) for all v e V if and only if x = y.(c) Let v1,...,vn be a basis for V. Prove that (x, v,) = (y, vj), i = 1,.... n, if and only if x = y.
(a) Prove the identity (u.v> = 1/4(||u +v||2 - ||u - v||2), (3.11)which allows one to reconstruct an inner product from its norm.(b) Use (3.11) to find the inner product on R2 corresponding to the norm||v|| = √u21 - 3v1v2 + 5v22.
(a) Show that, for all vectors x and y in an inner product space,||x + y||2 + ||x - y||2 = 2 (||x||2 + ||y||2).(b) Interpret this result pictorially for vectors in R2 under the Euclidean norm.
Suppose u. v satisfy ||u|| = 3, ||u + v|| = 4. and ||u - v|| = 6. What must ||v|| equal? Does your answer depend upon which norm is being used?
Let A be any n xn matrix. Prove that the dot product identity v • (Aw) = (ATv) • w is valid for any vectors v, w ∈ Rn.
Prove that A = AT is a symmetric n x n matrix if and only if {A v) • w = v • (A w) for all v. w ∈ Rn.
Prove that if (v, w) and «v, w» are two different inner products on the same vector space V. then their sum (((v, w))) = (v. w) + ((v. w)) defines an inner product on V.
Let V and VP be inner product spaces with respective inner products (v. v) and ((w, w)). Show that ((((v. w). (v,w)))) = (v. v) + «w, w)) for v, v ∈ V. w. w ∈ W defines an inner product on their Cartesian product V × W.
Let f(x) = x. g(x) = 1 + x2. Compute (f,g) ||f||. and ||g|| for(a) The L2 inner product(b) The L2 inner product(c) The weighted inner product
Which of the following formulas for (f, g) define inner products on the space C°[ - 1, 1 ]?a.b. c. d.
Prove thatdoes not define an inner product on the vector space C°[ - 1, 1]. Explain why this does not contradict the fact that it defines an inner product on the vector space C°[0, 1 ]. Does it define an inner product on the subspace P(n)
Does either of the following define an inner product on C°[0, 1 ]?(a) = f(0)g(0) + f(l)g(l)(b) (f.g) = f(0)g(0) + f(l)g(l) + f-10 f(x)g(x)dx
Let f(x) be a function, and ||f|| its L2 norm on [a,b]. Is ||f2|| = ||f||2? If yes, prove the statement. If no, give a counterexample.
Prove thatdefines an inner product on the space C'[a, b] of continuously differentiable functions on the interval [a,b]. Write out the corresponding norm, known as the Sobolev H' norm; it and its generalizations play an extremely important role in advanced mathematical analysis, [36],
Let V = C1 [ - 1, 1 ] denote the vector space of continuously differentiable functions for - 1 ‰¤ x ‰¤ 1.(a) Does the expression(b) Answer the same question for the subspace W = (f ˆˆ V| f(0) =0) consisting of all continuously differentiable functions which vanish at 0.
(a) Let h{x) ≥ 0 be a continuous, non-negative function defined on an interval [a. b]. Prove that ∫beh(x)dx = 0 if and only if h(x) = 0. Hint: Use the fact that ∫-di h(x)dx > 0 if h(x) > 0 for c
(a) Prove the inner product axioms for the weighted inner product (3.15), assuming w(x) ≤ 0 for all a ≤ x ≤ b.(b) Explain why it does not define an inner product if w is continuous and w(xo) < 0 for some x0 ∈ [a.b],(c) If w(.v) ≥ 0 for a ≤ x ≤ b. does (3.15) define an inner
Let Ω ⊂ R2 be a bounded domain. Let C°(Ω) denote the vector space consisting of all continuous, bounded real-valued functions f(x, y) defined for (x, y) ∈2.
Prove that each of the following formulas for (v. tv) defines an inner product on R3. Verify all the inner product axioms in careful detail:(a) V1W1 + 2V2w2 + 3v3w3(b) 4v1w1 + 2 v1w2 + 2v2wi +4v2W2 + V3W3(C) 2v1wi - 2v1w2 - 2V2W1 + 3v2w2 - v2w3 - v3w2 + 2v3w3
The unit circle for an inner product on R2 is defined as the set of all vectors of unit length: ||v|| = 1. Graph the unit circles for(a) the Euclidean inner product.(b) the weighted inner product (3.8).(c) the non-standard inner product (3.9).(d) Prove that cases (b). (c) are. in fact, both
(a) Explain why the formula for the Euclidean norm in R2 follows from the Pythagorean Theorem.(b) How do you use the Pythagorean Theorem to justify the formula for the Euclidean norm in R3? Hint: Look at Figure 3.1.
Prove that the norm on an inner product space satisfies ||cv|| = |c| ||v|| for any scalar c and vector v.
Prove that (av + bw + dw) = ac||v||2 + (ad + bc)(v. w) + 6d|w||2
Prove that the second bilinearity formula (3.4) is a consequence of the first and the other two inner product axioms.
Verify the Cauchy-Schwarz inequality for each of the following pairs of vectors v, w, using the standard dot product, and then determine the angle between them:(a) (1,2)T, (-1,2)T(b) (l,-1.0)T,(-1.0. l)T(c) (1,-1.0)T, (2,2.2)T(d) (1,-1,1,0)T, (-2,0. - 1, l)T(e) (2,1, -2, -l)T, (0, -1,2, - l)T
Show that one can determine the angle 6 between v and w via the formulaDraw a picture illustrating what is being measured.
The cross product of two vectors in R2 is defined as the scalar V x w = v1w2 - v2w1 (3.22) for v = (v1, v2)T, w = (w1, w2)T. (a) Does the cross product define an inner product on R2? Carefully explain which axioms are valid and which are not. (b) Prove that v x w = ||v|| ||w|| sin#, where 9 denotes
Verify the Cauchy-Schwarz inequality for the functions f(x) = x and g(x) = e' with respect to(a) the L2 inner product on the interval [0, 1 ],(b) the L2 inner product on [ - 1, 1 ],(c) the weighted inner product
Using the L2 inner product on the interval [0, nr] find the angle between the functions (a) 1 and cos x (b) 1 and sin x (c) cos x and sin x
Verify the Cauchy-Schwarz 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.
Find all vectors in R4 that are orthogonal to both (1,2, 3,4)T and (5,6,7,8)T.
(a) Find the Euclidean angle between the vectors (1, 1, 1, l)T and (1,1,1, -l)T inR4. (b) List the possible angles between (1, 1, 1, l)T and (a1, a2, a3, a4)T, where each a,- is either 1 or -1.
Find three vectors u. v and w in R3 such that u and v are orthogonal, u and w are orthogonal, but v and w are not orthogonal. Are your vectors linearly independent or linearly dependent? Can you find vectors of the opposite dependency satisfying the same conditions? Why or why not?
Prove that the only element w in an inner product space V that is orthogonal to every vector, so (w, v) = 0 for all v ∈ V. is the zero vector: w = 0.
A vector with ||v|| = 1 is known as a unit vector. Prove that if v. w are both unit vectors, then v + w and v - w are orthogonal. Are they also unit vectors?

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