New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
linear algebra

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

Letbe the augmented matrix for a linear system. For which values of a and b does the system have (i) A unique solution? (ii) Infinitely many solutions? (iii) No solution?
For which values of b and c does the system x1 + x2 + bx3 = 1, bx1 + 3x2 - x3 = -2, 3x1 + 4x2 + x3 = c, have (a) No solution? (b) Exactly one solution? (c) Infinitely many solutions?
Determine the general (complex) solution to the following systems: (a) 2x + (1 + i)y - 2iz = 2i (1 - i) x+y-2iz = 0 (b) x + 2 i y + (2 - 4 i )z = 5 + 5 i (-1 + i)x + 2y + (4 + 2i)z = 0 (1 - i)x + (l + 4i)y-5iz = 10 + 5i (c) x1 + 1x2 + x3 = 1 + 4 i -x1 +x2 - ix3 = -1 Ix1 - x2 - x3 = - 1 - 2i (d)
Write out a P A = LU factorization for each of the matrices in Exercise 1.8.7.(a)(b) (c) (d) (e) (f) (0 -1 2 5) (g) (h) (i)
Construct a system of three linear equations in three unknowns that has (a) One and only one solution; (b) More than one solution; (c) No solution.
Use Gaussian Elimination to find the determinant of the following matrices:(a)(b) (c) (d) (e) (f) (g)
(a) Prove that, for n odd, any n×n skew-symmetric matrix A = - AT is singular.(b) Find a nonsingular skew-symmetric matrix.
Prove directly that the 2 x 2 determinant formula (1.37) satisfies the four determinant axioms listed in Theorem 1.50.
In this exercise, we prove the determinantal product formula (1.82).(a) Prove that if E is any elementary matrix (of the appropriate size), thendet(E B) = det E det B.(b) Use induction to prove that ifA = E1 E2 ∙ ∙ ∙ Enis a product of elementary matrices, then det( AS) = det A det 5. Explain
Prove (1.83).
Prove (1.84). Use Exercise 1.6.28 in the regular case. Then extend to the nonsingular case. Finally, explain why the result also holds for singular matrices. det AT = det A. (1.84)
Write out the formula for a 4 x 4 determinant. It should contain 24 = 4! terms.
Show that (1.85) satisfies all four determinant axioms, and hence is the correct formula for a determinant.
Prove that axiom (iv) in Theorem 1.50 can be proved as a consequence of the first three axioms and the property det I =1.
Show that(a) ifis regular, then its pivots are a and det A/a: (b) if is regular, then its pivots are (c) Can you generalize this observation to regular n x n matrices?
Verify the determinant product formula (1.82) when
In this exercise, we justify the use of "elementary column operations" to compute determinants.Prove that(a) Adding a scalar multiple of one column to another does not change the determinant;(b) Multiplying a column by a scalar multiplies the determinant by the same scalar;(c) Interchanging two
Find the determinant of the Vandermonde matrices listed in Exercise 1.3.25. Can you guess the general n x n formula?
(a) Show that the nonsingular system ax + by = p, ex -I- dy = q has the solution given by the determinantal ratiosWhere (b) Use Cramer's Rule (1.87) to solve the systems (i) x + 3y=13. 4x + 2y = 0 (ii) x-2y = 4, 3x + 6y = - 2 (c) Prove that the solution to ax + by + cz = p. dx + ey + fz = q, gx
(a) Show that ifis a block diagonal matrix, where A and B are square matrices, then det D = det A det B. (b) Prove that the same holds for a block upper triangular matrix (c) Use this method to compute the determinant of the following matrices: (i) (ii) (iii) (iv)
(a) Give an example of a non-diagonal 2×2 matrix for which A2 = I. (b) If A2 = I, show that det A = ± 1
If A2 = A, what can you say about det A?
True or false: If true, explain why. If false, give an explicit counterexample.(a) If det ‰ 0 then A-l exists.(b) det(2A) = 2 det A.(c) det(A -(- B) = det A + det B.(d)(e) (f) det[(A + B)(A - B)] = det(A2 - B2). (g) If A is an n x n matrix with det A = 0, then rank A (h) If det A = I and
Prove that similar matrices have the same determinant: det A = det B whenever B = S-l A S.
Prove that if A is a n x n matrix and c is a scalar, then det(c A) = cn det A.
Prove that the determinant of a lower triangular matrix is the product of its diagonal entries.
Show that the set of complex numbers x + i y forms a real vector space under the operations of addition (x + iy) + (w + i v) = (x + u) + i (y + v) and scalar multiplication c(x + iy) = cx + i cy. (But complex multiplication is not a real vector space operation.)
The space R∞ is defined as the set of all infinite real sequences a = (a1, a2, a3,...) where ai ∈ R. Define addition and scalar multiplication in such a way as to make R∞ into a vector space. Explain why all the vector space axioms are valid.
Prove the basic vector space properties (i), (j), (k) following Definition 2.1.
Suppose that V and W are vector spaces. The Cartesian product space, denoted by V × W is defined as the set of all ordered pairs (v. w) where v ∈ V, w ∈ W, with vector addition (v, w) + ( , ) = (v + ,w + ) and scalar multiplication c(v, w) = (cv, cw).(a) Prove that V × W is a vector
Use Exercise 2.1.13 to show that the space of pairs (f(x), a) where f is a continuous scalar function and a is a real number forms a vector space. What is the zero element? Be precise! Write out the laws of vector addition and scalar multiplication.
Show that the positive quadrant Q = {(x,y) | x, y > 0) ⊂ R2 forms a vector space if we define addition by (x1, y1) + (x2, > y2) = (x1 x2, y1 y2) and scalar multiplication by c (x, y) = (xc, yc).
Let S be any set. Carefully justify the validity of all the vector space axioms for the space T(S) consisting of all real-valued functions f: S → R.
Let F(R2, R2) denote the vector space consisting of all functions f: R2 †’ R2.(a) Which of the following functions f(x, y) are elements?(i) x2 + y2(ii)(iii) (iv) (v) (vi) (b) Sum all of the elements of ,F(R2, R2) you identified in part (a). Then multiply your sum by the scalar -5. (c)
A planar vector field is a function which assigns a vectorR2. Explain why the set of all planar vector fields forms a vector space.
Let h, k > 0 be fixed. LetS = {(ih, jk) | 1 ‰¤ I ‰¤ m, I ‰¤ j ‰¤ n)be points in a rectangular planar grid. Show that the function space T(S) can be identified with the vector space of m Ã— n matrices MmÃ—n.
(a) Prove that the set of all vectors (x, y,z)T such that x -y + 4z = 0 forms a subspace of R3.(b) Explain why the set of all vectors that satisfy x - y + 4z = 1 does not form a subspace.
The trace of an n × n matrix A ∈ Mn×n is defined to be the sum of its diagonal entries: tr A = a11 +a22 H+. . . + ann. Prove that the set of trace zero matrices, tr A = 0, is a subspace of Mm×n?
(a) Is the set of n x n matrices with det A = 1 a subspace of Mn×n? (b) What about the matrices with det A = 0?
Which of the following are vector spaces? Justify your answer! (a) The set of all row vectors of the form (a, 3a). (b) The set of all vectors of the form (a, a + 1). (c) The set of all continuous functions for which f(- 1)= 0. (d) The set of all periodic functions of period 1, i.e., f(x + 1) =
Let V = Cn(R) be the vector space consisting of all continuous functions f: R → R. Explain why the set of all functions such that f(1) = 0 forms a subspace, but the set of functions such that f(0) = 1 does not. For which values of a, b does the set of functions such that f(a) = b form a subspace?
Let V = C ([a, b|) be the vector space consisting of all functions f(t) which are defined and continuous on the interval 0 ‰¤ t ‰¤ 1. Which of the following conditions define subspaces of V? Explain your answer.(a) f(0) = 0(b) f(0) = 2/(l)(c) f(0)f(l)= 1(d) f(0) = 0 or f(l)
Prove that the set of solutions to the second order ordinary differential equation u" = xu forms a vector space.
(a) Prove that C1 ([a, b], R2), which is the space of continuously differentiable parametrized plane curves f: [a, b] → R2, forms a vector space.(b) Is the subset consisting of all curves that go through the origin a subspace ?
A planar vector field v(.x, y) = (u(x, y), v(x, y))T is called irrotational if it has zero divergence:Prove that the set of all irrotational vector fields forms a subspace of the space of all vector fields.
Show that if w and Z are subspaces of V. then(a) Their intersection W ∩ Z is a subspace of V,(b) Their sum W + Z = { w + z | w ∈ W. z ∈ Z ) is also a subspace, but(c) Their union WUZ is not a subspace of V, unless W ⊂ Z or Z ⊂ W.
Let V be a vector space. Prove that the intersection ∩ of any collection (finite or infinite) of subspaces W, ⊂ V is a subspace.
Let W ⊂ V be a subspace. A subspace Z ⊂ V is called a complementary subspace to W if(i) W ∩ Z = {0}, and(ii) W + Z = V, i .e., every v ∈ V can be written as v = w + z for w ∈ W and z ∈ Z.(a) Show that the .v and y axes are complementary subspaces of R2.(b) Show that the lines x = y and
(a) Show that V0 = ( (v, 0) | v ∈ V ) and w0 = ( (0, w) | w e w ) are complementary subspaces, as in Exercise 2.2.24, of the Cartesian product space V × W, as defined in Exercise 2.1.13.(b) Prove that the diagonal D = {(v. v)) and the amidiagonal A = {(v, -v)) are complementary subspaces of V ×
(a) Show that the set of even functions, f(-x) = f(x) is a subspace of the vector space of all functions .F(R). (b) Show that the set of odd functions g(- x) = - g(x) forms a complementary subspace, as defined in Exercise 2.2.24. (c) Explain why every function can be uniquely written as the sum of
Show that the subspace of skew-symmetric n × n matrices forms a complementary subspace to the space of symmetric n × n matrices. Explain why this implies that every square matrix can be uniquely written as the sum of a symmetric and a skew-symmetric matrix.
Define(a) Prove that all derivatives of / vanish at the origin: f(n) (0) = 0 for n =0, 1,2,. . . . (b) Prove that f(x) is not analytic by showing that its Taylor series at a = 0 does not converge to f(x) when x > 0.
Let(a) Find the Taylor series of f at a = 0. (b) Prove that the Taylor series converges for | x | (c) Prove that f(x) is analytic at x = 0.
Graph the following subsets of R3 and use this to explain which are subspaces: (a) The line (t. -3t)T for t ∈ R. (b) The helix (cost, sin t, t)T. (c) The surface x - 2y + 3z = 0. (d) The unit ball x2 + y2 + z2 < 1. (e) The cylinder (y + 2)2 -f (z - 1 )2 = 5. (f) The intersection of the
Let V be a vector space. A subset of the form A = {w + a | w ∈ W} where V ⊂ V is a subspace and a ∈ V is a fixed vector is known as an affine subspace of V.(a) Show that the affine subspace A ⊂ P is a genuine subspace if and only if a ∈ W.(b) Draw the affine subspaces A ⊂ R2 when(i) W
Show that if W ⊂ R3 is a subspace containing the vectors (1,2, - l)T, (2,0. l)T, (0,-1,3)T, then W = R3.
(a) Can you construct an example of a subset S ⊂ R2 with the property that eve S for any c ∈ R, v ∈ S, and yet S is not a subspace? (b) What about an example where v + w ∈ S for every v, w ∈ S, and yet S is not a subspace?
Prove that the set of all solutions x of the linear system A x = b forms a subspace if and only if the system is homogeneous
A square matrix is called strictly lower triangular if all entries on or above the main diagonal are 0. Prove that the space of strictly lower triangular matrices forms a subspace of the vector space of all n × n matrices.
Show thatbelongs to the subspace of R3 spanned by by writing it as a linear combination of the spanning vectors
Write the following trigonometric functions in phase-amplitude form: (a) Sin3x, (b) Cosx - sinx (c) 3cos2x + 4sin2x (d) Cosxsinx
(a) Prove that the set of solutions to the homogeneous ordinary differential equation uʹʹ - 4uʹ + 3 u = 0 forms a vector space. (b) Write the solution space as the span of a finite number of functions. (c) What is the minimal number of functions needed to span the solution space?
Find a finite set of real functions that spans the solution space to the following homogeneous ordinary differential equations: (a) uʹ - 2u = 0 (b) uʹʹ+ 4u = 0 (c) uʹʹ - 3uʹʹ= 0 (d) uʹʹ + 4uʹ+3u =0 (e) uʹʹ + uʹ + u = 0 (f) uʹʹʹ - 5uʹʹ = 0 (g) u(4) + u = 0
Consider the boundary value problem uʹʹ +4u = 0, 0 ≤ x ≤ π, u(0) = 0, u(π) = 0. (a) Prove, without solving, that the set of solutions forms a vector space. (b) Write this space as the span of one or more functions.
Which of the following functions lie in the span of the vector-valued functions(a)(b)(c)(d)(e)
Prove or give a counter-example: if z is a linear combination of u, v, w. then w is a linear combination of u, v, z.
Suppose v1... ,vm span V. Let vm+1,... ,vn ∈ V be any other elements. Prove that the combined collection v1.......vn also spans V.
(a) Show that if v is a linear combination of v1.......vm, and each vj is a linear combination of w1,..., wm then v is a linear combination of w1..........wn. (b) Suppose v1.....vm, span V. Let w1,..., wn ∈ V be any other elements. Suppose that each vi can be written as a linear combination of
Show thatis in the subspace of R4 spanned
The span of an infinite collection v1, v2, v3,... ˆˆ V of vector space elements is defined as the set of all finite linear cominationsis finite but arbitrary. (a) Prove that the span defines a subspace of the vector space V. (b) What is the span of the monomials 1, x, x2, x3. . . ?
Determine whether the given vectors are linearly independent or linearly dependent:(a)(b) (c) (d) (e) (f) (g) (h) (i)
(a) Show that the vectorsare linearly independent. (b) Which of the following vectors are in their span? (i) (ii) (iii) (iv) (c) Suppose b = (a,b,c,d)T lies in their span. What conditions must a, b, c, d satisfy?
(a) Show that the vectorsare linearly independent. (b) Show that they also span R4. (c) Write (1.0. 0, I )T as a linear combination of them.
Determine whether the given row vectors are linearly independent or linearly dependent: (a) (2, 1), ( - 1,3), (5,2) (b) (1,2,-1), (2, 4, -2) (c) (1,2,3),(1,4,8),(1,5.7) (d) (1, 1,0), (1,0. 3). (2.2, 1), (1.3,4) (e) (1,2, 0, 3), (-3, -1,2, -2), (3, -4, -4, 5) (0 (2,1,-1,3), (-1,3,1,0), (5,1,2,-3)
Let x and y be linearly independent elements of a vector space V. Show that u = ax + by, and v = cx + dy are linearly independent if and only if ad - bc ≠ 0. Is the entire collection x. y, u, v linearly independent?
(a) Determine whetheris in the span of (b) Is In the span of and (c) Is in the span of and
Prove parts (b) and (c) of Theorem 2.21. Theorem 2.21 (b) The vectors are linearly independent if and only if the only solution to the homogeneous system Ac = 0 is the trivial one, c = 0. (c) A vector b lies in the span of v1,..., vk if and only if the linear system Ac = b is compatible, i.e., has
(a) Prove that if v1,.......vn are linearly independent, then any subset, e.g., v1.......vk with k < m, is also linearly independent. (b) Does the same hold true for linearly dependent vectors? Prove or give a counterexample.
Determine whether the given functions are linearly independent or linearly dependent: (a) 2 - x2, 3x, x2 + x - 2 (b) 3x - 1, a (2 a + 1), jc (a- - 1) (c) ex, ex+1 (d) sin x. sin(A + 1) (e) ex,ex+l,ex+2 (0 sinx, sin(x-+1). sin(x + 2) (g) ex,xex,x2ex (h) ex, e2x. e3x (i) x+.y, x-y+1, a+3y+2 -these
Show that the functions f(x) = x and g(x) = |x| are linearly independent when considered as functions on all of R, but are linearly dependent when considered as functions defined only on R+ = (U > 0).
(a) Prove that the polynomialsi = 1.........k, are linearly independent if and only if the k Ã— (n + 1) matrix A whose entries are their coefficients aij ,1 ‰¤ i ‰¤ k, 0 ‰¤ j ‰¤ n. has rank k.(b) Formulate a similar matrix condition for testing whether another polynomial q(x) lies
The Fundamental Theorem of Algebra, [22], states that a non-zero polynomial of degree n has at most n distinct real roots p(x) = 0. Use this fact to prove linear independence of the monomial functions 1,x,x2. . . , xn.
Let x1, x2,. . . , xn be a distinct set of sample points. (a) Prove that the functions f1(x),... , fk(x) are linearly independent if their sample vectors f1......... fλ are linearly independent vectors in Rn. (b) Give an example of linearly independent functions that have linearly dependent
Suppose f1(t).........fk(t) are vector-valued functions from R to Rn.(a) Prove that if f1(t0),. . . fk(t0) afe linearly independent vectors in Rn at one fixed to, then f1 (t),. . . fk(t) are linearly independent functions.(b) Show thatare linearly independent functions, even though at each fixed
The Wronskian of a pair of differentiable functions f(x). g(x) is the scalar function(a) Prove that if /. g are linearly dependent, then W[f(x). g(x)] = 0.(b) Prove that if W[f(x). g(x) ‰ 0, then f, g are linearly independent.(c) Let f(x) = x3, g(x) = |x|3. Prove that f. g ˆˆ C2 are twice
(a) Graph the subspace of R3 spanned by the vector v, = (3.0. l)T.(b) Graph the subspace spanned by the vectors v1 = (3,-2,-1)T. v2 = (-2,0.-1)T.(c) Graph the span of v1 = (1,0, - l)T, v2 = (0,-l, l)T,v3 = (l,-1.0)T.
Let U be the subspace of R3 spanned by u, = (1, 2, 3)T, u2 = (2, -1, 0)T. Let V be the subspace spanned by v1 = (5.0.3)T, v2 = (3, 1.3)T. Is V a subspace of U? Are U and V the same?
(a) Let S be the subspace of M2Ã—2 consisting of all symmetric 2Ã—2 matrices. Show that S is spanned by the matricesand
(a) Determine whether the polynomials x2 + 1, x2 - 1, x2 + x + I span P(2). (b) Do x3 - I, x2 + 1, x - 1, 1 span P(3)? (c) What about x3, x2 + 1, x2 - x, x + 1?
Which of the following functions lies in the subspace spanned by the functions 1, x, sinx and sin2 x? (a) 3 - 5x (b) x2 + sin2 x (c) sinx -2cosx (d) cos2x (e) xsinx (f) ex
(a) Prove that 1 +t2,1 +t2,1 + 2t +t2 is a basis for the space of quadratic polynomials P(2). (b) Find the coordinates of p(t) = 1 + 4t + 7t2 in this basis.
(a) Show that 1. (1- t), (1 - t)2, (1- t)3 is a basis for P(3). (b) Write p(t) = 1 + t3 in terms of the basis elements.
Let P(4) denote the vector space consisting of all polynomials p(x) of degree ≤ 4. (a) Are p1(x) = x3 - 3x + l. p2(x) = x4 - 6x + 3. P3(x) = x4 - 2x3 + 1, linearly independent elements of P(4)? (b) What is the dimension of the subspace of P(4) spanned by p1, p2, p3?
Let(a) Show that the sample vectors corresponding to the functions 1. cos Ï€ x. cos2 Ï€ x and cos 3 Ï€ x form a basis for the vector space of all sample functions on S. (b) Write the sampled version of the function f(x) = x in terms of this basis.
(a) Prove that the vector space of all 2 × 2 matrices is a four-dimensional vector space by exhibiting a basis.(b) Generalize your result and prove that the vector space Mm×m consisting of all m × n matrices has dimension mn.
(a) Find a basis for and the dimension of the space of upper triangular 2×2 matrices. (b) Can you generalize your result to upper triangular n x n matrices?
(a) What is the dimension of the vector space of 2 × 2 symmetric matrices? (b) Of skew-symmetric matrices? (c) Generalize to 3 × 3 case. (d) What about n × n matrices?
A matrix is said to be a semi-magic square if its row- sums and column sums (i.e., the sum of entries in an individual row or column) all add up to the same number. An example iswhose row and column sums are all equal to 15.(a) Explain why the set of all semi-magic squares forms a subspace.(b)
Show, by example, how the uniqueness result in Lemma 2.34 fails if one has a linearly dependent set of vectors.
Suppose that v1,... ,vn form a basis for Rn. Let A be a nonsingular matrix. Prove that Av1........Avn also form a basis for Rn. What is this basis if you start with the standard basis: vi = ei?

Showing 7900 - 8000 of 11883

First
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved