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

Elementary Linear Algebra with Applications 9th edition Howard Anton, Chris Rorres - Solutions

Assume that v1, v2, and v3 are vectors in R3 that have their initial points at the origin. In each part, determine whether the three vectors lie in a plane. v1 = (2, -2, 0) v2 = (6, 1, 4) v3 = (2, 0, -4)
For which real values of Î» do the following vectors form a linearly dependent set in R3?
Determine the dimension of and a basis for the solution space of the system. x1 - 3x2 + x3 = 0 2x1 - 6x2 + 2x3 = 0 3x1 - 9x2 + 3x3 = 0
Find a standard basis vector that can be added to the set (v1, v2) to produce a basis for R3. v1 = (- 1, 2, 3), v2 = (1, - 2, - 2)
Let {v1, v2, v3} be a basis for a vector space V. Show that {u1, u2, u3} is also a basis, where u1 = v1, u2 = v1 + v2, and u3 = v1 + v2 + v3.
Let S be a basis for an n-dimensional vector space V. Show that if v1, v2, ... vr form a linearly independent set of vectors in V, then the coordinate vectors (v1)s, (v2)s, (vr)s form a linearly independent set in Rn, and conversely.
Find a basis for the subspace of P2, spanned by the given vectors. - 1 + x - 2x2, 3 + 3x + 6x2, 9
Which of the following sets of vectors are bases for R3? (a) (1, 0, 0), (2, 2, 0), (3, 3, 3) (b) (2, 3, 1), (4, 1, 1), (0, - 7, 1)
The basis that we gave for M22 m Example 6 consisted of noninvertible matrices. Do you think that there is a basis for M22 consisting of invertible matrices? Justify your answer.
Confirm your conjecture by finding a basis. The equation x1 + x2 + ... xn = 0 can be viewed as a linear system of one equation in n unknowns.
Show that the following set of vectors is a basis for M22.
Find the coordinate vector of w relative to the basis S = (u1, u2) for R2 (a) u1 = (1, 0), u2 = (0, 1); w = (3, - 7) (b) u1 = (2, - 4) , u2 = (3, 8 ); w = (1 , 1)
Find the coordinate vector of v relative to the basis S = (v1, v2, v3) v = (2, - 1, 3), v1 = (1, 0, 0), v2(2, 2, 0), v3 = (3, 3, 3)
Find a basis for the subspace of R4 spanned by the given vectors. (1, 1, - 4, - 3), (2, 0, 2, -2), (2, - 1, 3, 2)
Prove that the row vectors of an n × n invertible matrix A form a basis for Rn.
Find a 3 × 3 matrix whose nullspace is (a) A point (b) A line (c) A plane
(a) Find all 2 Ã— 2 matrices whose nullspace is the line 3x - 5y = 0(b) Sketch the nullspaces of the following matrices:
Suppose that A and B are n × n matrices and A is invertible. Invent and prove a theorem that describes how the row spaces of AB and B are related.
Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of(a)(b) (c)
Find the vector form of the general solution of the given linear system Ax = 1; then use that result to find the vector form of the general solution of Ax = 0 (a) x1 - 3x2 = 1 2x1 - 6x2 = 2 (b) x1 - 2x2 + x3 + 2x4 = - 1 2x1 - 4x2 + 2x3 + 4x4 = - 2 3x1 - 6x2 + 3x3 + 6x4 = - 3
For the matrices in Exercise 6, find a basis for the column space of A.(a)(b)
Suppose that A is a 3 × 3 matrix whose nullspace is a line through the origin in 3-space. Can the row or column space of A also be a line through the origin? Explain.
Are there values of r and s for which
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) If A is not square, then the row vectors of A must be linearly dependent. (b) If the row vectors and the column vectors of A are linearly independent, then A
In each part, use the information in the table to determine whether the linear system Ax = b is consistent. If so, state the number of parameters in its general solution.
What conditions must be satisfied by b1, b2, b3, b4 and b5 for the over determined linear system x1 - 3x2 = b1 x1 - 2x2 = b2 x1 + x2 = b3 x1 - 4x2 = b4 x1 + 5x2 = b5 to be consistent?
In each part, the solution space is a subspace of R3 and so must be a line through the origin, a plane through the origin, all of R3 or the origin only. For each system, determine which the case is. If the subspace is a plane, find an equation for it, and if it is a line, find parametric
In advanced linear algebra, one proves the following determinant criterion for rank: The rank of a matrix A is r if and only if A has some r Ã— r submatrix with a nonzero determinant, and all square submatrices of larger size have determinant zero. (A submatrix of A is any matrix
Prove: If S is a basis for a vector space V, then for any vectors u and v in V and any scalar k, the following relationships hold: (ku)S = k(u)S
(a) Express v = (1, 1) as a linear combination of v1 = (1, - 1), = (3, 0), and v3 = (2, 1) in two different ways. (b) Explain why this does not violate Theorem 5.4.1.
Let (u, v) be the Euclidean inner product on R2, and let u = (3, - 2), v = (4, 5), w = (-1, 6), and k = -4. Verify that (a) (u, v + w) = (u, v) + {u, w) (b) (ku, v) = k{u, v) = {u, kv)
Use the inner products in Exercise 10 to find d (u, v) for u = ( - 1, 2) and v = (2, 5).Inn Products From Exercise 10(a) The weighted Euclidean inner product (u, v) = 3u1v1 I 2u2v2> where u = (u1, u2) and v = (v1, v2)(b) The inner product generated by the matrix
Let M22 have the inner product in Example 7. Find d(A, B).
Let the vector space P2 have the inner product(a) Find ||p|| for p = 1, = p = x2. (b) Find d(p, q) if p = 1 and q = x.
Let p = p(x) and q = q(x) be polynomials in p2. Show thatis an inner product on p2. Is this an inner product on P3? Explain.
Use the inner productto compute {p, q), for the vectors p = p(x) and q = q(x) in p3. p = x-5x3, q = 2 + 8x2
Show that the inner product in Example 7 can be written as (U,V) = tr (UTV).
Show that matrix 5 generates the weighted Euclidean inner product (u, v) = w1u1v1 + w2u2v2+ ∙ ∙ ∙ +wnunvn,.
Prove parts (a) and (d), of Theorem 6.1.1, justifying each step with the name of a vector space axiom or by referring to previously established results. Theorem 6.1.1 If u, v and w are vectors in a real inner product space, and k is any scalar, then (a) (0, v) = (v,0) = 0 (d) (u - v, w ) = (u,w) +
(a) Use Formula 3 to show that (u, v) = 9u1v1 + 4u2v2 the inner product on pl generated by(b) Use the inner product in part (a) to compute (u, v) if u = (-3, 2) and v = (1, 7).
Let u = (u1, u2, u3) and v = (v1, v2, v3). Determine which of the following inner products on p are}. For those that are not, list the axioms that do not hold. (a) (u, v) = u21v21 + u22v22 + u23v23 (b) (u, v) = 2u1v1 + u2v2 + 4u3v3
Let R4 have the Euclidean inner product. Find two unit vectors that are orthogonal to the three vectors u = (2, 1, -4, 0), v = ( - 1, -1,2, 2), and w = (3, 2, 5,4).
In each part, verify that the Cauchy-Schwarz inequality holds for the given vectors. u = (- 2, 1) and y = (1, 0) using the inner product of Example 2 of Section 6.1
Let W┵ the plane in R3 with equation x - 2y - 3z = 0- Find parametric equations for W
Let A be the matrix in Exercise 16. Find bases for the column space of A and nullspace of AT
Let V be an inner product space. Show that if u and v are orthogonal unit vectors in V, then ||u - v|| = √2.
Let V be an inner product space. Show that if w is orthogonal to each of the vectors u1,u2, ... ,ur, then it is orthogonal to every vector in span (u1, u2,..., ur,}.
Let {w1, w2,..., wk} be a basis for a subspace W of V. Show that W⊥ consists of all vectors in V that are orthogonal to every basis vector.
Prove the following parts of Theorem 6.2.2: Part (c)
Let R3 have the Euclidean inner product. Let u = (l, 1, -1) and y = (6, 7, -15). If ||ku + v|| = 13, what is K?
Use vector methods to prove that a triangle that is inscribed in a circle so that it has a diameter for a side must be a right triangle.
Let f (x) and g(x) be continuous functions on [0, 1], Prove:(a)(b)
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) ||u + v + w|| ≤ ||u|| + ||v|| + ||w|| for all vectors u, v, and w in an inner product space. (b) If u is in the row space and the nullspace of a square
Let R2, R3, and R4 have the Euclidean inner product. In each part, find the cosine of the angle between u and v. (a) U= (1, - 3), v = (2, 4) (b) U = (- 1, 5, 2), v= (2, 4, -9) (c) U= (1, 0, 1, 0), v = (-3, -3, -3, -3)
LetWhich of the following matrices are orthogonal to A with respect to the inner product in Exercise 8? (a) (b)
In each part, an orthonormal basis relative to the Euclidean inner product is given. Use Theorem 6.3.1 to find the coordinate vector of w with respect to that basis.
Let R3 have p3 have the Euclidean inner product. Use the Gram-Schmidt process to transform the basis (u1, u2,u3) into an orthonormal basis. u1 = (1, 1, 1), u2 = ( - 1, 1, 0), u3 = (1, 2, 1)
Let R3 have the Euclidean inner product. Find an orthonormal basis for the subspace spanned by (0, 1, 2), (-1, 0, 1), (-1, 1, 3).
The subspace of R3 spanned by the vectorsand u2 = (0, 1, 0) is a plane passing through the origin. Express w = (1, 2, 3) in the form w = w1 + w2, where w1 lies in the plane and w2 is perpendicular to the plane.
Let (v1, v2, v3) be an orthonormal basis for an inner product space V. Show that if is a vector in V, then ||w||2 = (w, v1)2 + (w, v2)2 + (w, v3)2.
In Step 3 of the proof of Theorem 6.3.6, it was stated that "the linear independence of {u1,u2,. . . . un) ensures that v3 ≠0. " Prove this statement.
Let the vector space p2 have the inner productApply the Gram-Schmidt process to transform the standard basis S= {1, x, x2} into an orthonormal basis. (The polynomials in the resulting basis are called the first three normalized Legendre polynomials.)
Let p2 have the inner productApply the Gram-Schmidt process to transform the standard basis S= {1, x, x3} into an orthonormal basis.
Prove Theorem 6.3.5.Let W be a finite-dimensional subspace of an inner product space V.If (v1, v2,. . . . vr) is an orthonormal basis for W, and u is any vector in V, then
Find vectors x and y in R2 that are orthonormal with respect to the inner product (u, v) = 3u1v1 + 2u2v2 but are not orthonormal with respect to the Euclidean inner product.
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) A linearly dependent set of vectors in an inner product space cannot be orthonormal. (b) Every finite-dimensional vector space has an orthonormal basis. (c)
Verify that the given set of vectors is orthogonal with respect to the Euclidean inner product; then convert it to an orthonormal set by normalizing the vectors. (1, 0, -1), (2, 0,2), (0,5,0)
Verify that the vectorsForm an orthonormal basis for R3 with the Euclidean inner product; then use Theorem 6.3.1 to express each of the following as linear combinations of v1, v2, and v3. (3, -7, 4)
Find the normal system associated with the given linear system.
Let W be the line with parametric equations x = 2t, y = - t, z = 4t (- ∞ < t < ∞) (a) Find a basis for W. (b) Use Formula 6 to find the standard matrix for the orthogonal projection onto W. (c) Use the matrix obtained in (b) to find the orthogonal projection of a point P0(x0,y0,z0- zo) onto
For the linear systems in Exercise 3, verify that the error vector Ax - b resulting from the least squares solution x is orthogonal to the column space of A.(a)(b)
Let A be an m × n matrix with linearly independent row vectors. Find a standard matrix for the orthogonal projection of Rn onto the row space of A.
Repeat Exercise 18 under the assumption that the relationship between the current I and the voltage drop V is best modeled by an equation of the form V = IR + c, where c is a constant offset value. This leads to a 5 × 2 linear system. Exercise 18 Successive experiments are performed in which a
Find the least squares solution of the linear system Ax = b, and find the orthogonal projection of b onto the column spaces o A.(a) (b)
Find the orthogonal projection of u onto the subspace of p4 spanned by the vectors v1, v2, and v3. u = (6, 3, 9, 6); v1 = (2, 1, 1, 1), v2 = (1, 0, 1, 1), v3 = (-2, - 1, 0, - 1)
Use Formula 6 and the method of Example 3 to find the standard matrix for the orthogonal projection p: R2 → R2 onto The x-axis
Find the coordinate vector for w relative to the basis S= {u1, u2} for R2. U1 = (2, - 4), u2 = (3, 8); w = (1, 1)
Let V be the space spanned by f1 = sin x and f2 = cos π- (a) Show that gl = 2 sin x + cos x and g2 = 3 cos x form a basis for V. (b) Find the transition matrix from B' = {g1, g2) to B = (f1, f2}. (c) Find the transition matrix from B to Bʹ (d) Compute the coordinate vector [h]B, where h = 2 sin x
Find the coordinate vector for p relative to S = {P1, P2, P3} p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2
Consider the coordinate vectors(a) Find w if S is the basis in Exercise 2(a). (b) Find B if S is the basis in Exercise 4.
Repeat the directions of Exercise 6 with the same vector w but withExercise 6 Consider the bases B = {u1, u2} and BÊ¹ = {v1, v2} for R2, where (a) Find the transition matrix from B' to B. (b) Find the transition matrix from B to BÊ¹. (c) Compute the coordinate vector [w]B,
What conditions must a and b satisfy for the matrixto be orthogonal?
Use the result in Exercise 14 to prove that multiplication by a 2 × 2 orthogonal matrix is either a rotation or a rotation followed by a reflection about the x-axis.
Prove the equivalence of statements (a) and (c) in Theorem 6.6.1. Theorem 6.6.1 (a) A is orthogonal. (c) The column vectors of A form an orthonormal set in Rn with the Euclidean inner product.
Referring to Exercise 20, what can you say about det (A) if A is the standard matrix for a rigid linear operator on R2?
Determine which of the following matrices are orthogonal. For those that are orthogonal, find the inverse.(a)(b) (c)
Repeat Exercise 8 for a rotation of θ = π/3 counterclockwise about the y-axis (looking along the positive y-axis toward the origin) Exercise 8 Let a rectangular xʹyʹzʹ -coordinate system be obtained by rotating a rectangular xyz-coordinale system counterclockwise about the z-axis (looking down
Let R4 have the Euclidean inner product (a) Find a vector in R4 that is orthogonal to u1 = (1,0,0,0) and u4 = (0,0,0,1) and makes equal angles with u2 = (0,1,0,0) and u3 = (0,0,1,0). (b) Find a vector x = (x1,x2, x3, x4) of length 1 that is orthogonal to u1 and u4 above and such that the cosine of
(a) In R3 the vectors (k, 0, 0), (0, k, 0), and (0, 0, k) form the edges of a cube with diagonal (k, k, k) (Figure 3.3.4). Similarly, in Rn the vectors (k, 0,0,..., 0), (0, k, 0......0)........ (0,0,0,.., k) can be regarded as edges of a "cube" with diagonal (k, k, k,..., k). Show that each of the
Let u be a vector in an inner product space V, and let (v1, v2, ..., vn) be an orthonormal basis for V. Show that if a1 is the angle between u and vi then Cos2a1 + cos2α2 + . . . + cos2an = 1
Prove part (c) of Theorem 6.2.5. Theorem 6.2.5. (c) The orthogonal complement of W⊥ is W; that is, (W⊥) ⊥ = W.
Find a weighted Euclidean inner product on Rn such that the vectors v1 = (1, 0, 0,..., 0) v2 = (0, √2,0,..., 0) v3 = (0, 0, √3,..., 0) . . . vn = (0,0,0, . . .√n) Form an orthonormal set.
Prove: If O is an orthogonal matrix, then each entry of O is the same as its cofactor if det(Q) = 1 and is the negative of its cofactor if det (Q)= -1.
Find the characteristic equations of the following matrices:(a)(b)
Find the eigenvalues of A9 for
Let A be a 2 Ã— 2 matrix, and call a line through the origin of R2 invariant under A if Ax lies on the line when x does. Find equations for all lines in R2, if any, that are invariant under the given matrix.(a)(b)
Let A be an n × n matrix. Prove that the characteristic polynomial of A has degree n. Prove that the coefficient of λn in the characteristic polynomial is 1.
Use the result in Exercise 16 to show that ifthen the solutions of the characteristic equation of A are Use this result to show that A has (a) Two distinct real eigenvalues if (a - d)2 + 4bc > 0 (b) Two repeated real eigenvalues if (a - d)2 + 4bc = 0 (c) Complex conjugate eigenvalues if (a - d)2 +
Prove: If a, b, c, and d are integers such that a + b = c + d, thenhas integer eigenvalues-namely, Î»j = a + b and Î»2 = a - c.
Prove: If A is an eigenvalue of A, x is a corresponding eigenvector, and 5 is a scalar, then λ -s is an eigenvalue of A - sI, -and x is a corresponding eigenvector.

Showing 7400 - 7500 of 11883

First
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
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