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linear algebra

Elementary Linear Algebra with Applications 9th edition Bernard Kolman, David Hill - Solutions

Let W and U be subspaces of vector space V. (a) Show that W ( U, the set of all vectors v that are either in W or in U, is not always a subspace of V. (b) When is W ( U a subspace of V? (c) Show that W ( U, the set of all vectors v that are in both W and U, is a subspace of V.
Prove that a subspace W of R3 coincides with R3 if and only if it contains the vectorsAnd
Let W be a nonempty subset of a vector space V. Prove that W is a subspace of V if and only if ru + sv is in W for any vectors u and v in W and any scalars r and s.
Let V be the set of all polynomials of (exactly) degree 2 with the definitions of addition and scalar multiplication as in Example 6. (a) Show that V is not closed under addition. (b) Is V closed under scalar multiplication? Explain.
Let V be the set of all 2 x 2 matrices(a) Is V closed under addition?(b) Is V closed under scalar multiplication?(c) What is the zero vector in the set V?(d) Does every matrix A in V have a negative that is in V? Explain.(e) Is V a vector space? Explain.
Prove in detail that Rn is a vector space?
Show that P, the set of all polynomials, is a vector space?
Let W be the set of all 3 ( 3 matrices of the formShow that W is a subspace of M33.
Let W be the set of all 2 ( 2 matricesSuch that Az = 0, where Is W a subspace of M22? Explain.
Show that every vector in R3 of the formFor t any real number, is in the null space of the matrix
Let A be an m ( n matrix. I s the set W of all vectors x in Rn such that Ax ( 0 a subspace of Rn ? Justify your answer
Show that the only subspaces of R1 are {0} and R1 itself.
Show that the only subspaces of R2 are {0}, R2, and any set consisting of all scalar multiples of a nonzero vector.
The set W of all 2 ( 3 matrices of the formWhere c = a + b, is a subspace of M23. Show that every vector in W is a linear combination of And
The set W of all 2 ( 2 symmetric matrices is a subspace of M22. Find three 2 ( 2 matrices v1, v2, and v3 so that every vector in W can be expressed as a linear combination of v1, v2, and v3.
(a) Show that a line t 0 through the origin of R3 is a subspace of R3. (b) Show that a line t in R3 not passing through the origin is not a subspace of R3.
The set W of all 2 ( 2 matrices A with trace equal to zero is a subspace of M22. LetShow that span S = W.
The set W of all 3 ( 3 matrices A with trace equal to zero is a subspace of M33. Determine a subset S of W so that span S = W.
Let T be the set of all matrices of the form AB - BA, where A and B are n ( n matrices. Show that span T is not Mnn.
In each part, explain why the set S is not a spanning set for the vector space V.(a) S = {t3, t2, t} V = P3(b)(c)
Let u and v be nonzero vectors in a vector space V. Show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other.
Let S = {v1, v2, v3} be a set of vectors in a vector space V. Prove that 5 is linearly dependent if and only if one of the vectors in 5 is a linear combination of all the other vectors in 5.
Suppose that S = {v1, v2, v3} is a linearly independent set of vectors in a vector space V. Prove that T = {w1, w2, w3} is also linearly independent, where w1 = v1 + v2 + v3, w2 = v2 + v3, and w3 = v3.
Suppose that 5 = {v1, v2, v3} is a linearly independent set of vectors in a vector space V. Is T = {w1, w2, w3}, where w1 = v1 + v2, w2 = v1 + v3, w3 = v2 + v3, linearly dependent or linearly independent? Justify your answer.
Suppose that S = [v1, v2, v3} is a linearly dependent set of vectors in a vector space V. Is T = {w1, w2, w3}, where w1 = v1, w2 = v1, + v3, w3 = v1, + v2 + v3, linearly dependent or linearly independent? Justify your answer.
Show that if {v1, v2) is linearly independent and v3 does not belong to span {v1, v2}, then {v1, v2, v3} is linearly independent.
Suppose that {v1, v2, vn} is a linearly independent set of vectors in Rn. Show that if A is an n ( n non-singular matrix, then {Av1, Av2, Ayn} is linearly independent.
Let A be an m ( n matrix in reduced row echelon form. Prove that the nonzero rows of A, viewed as vectors in Rn, form a linearly independent set of vectors.
Let S = {u1, u2, ( ( ( ( uk} be a set of vectors in a vector space and let T = {v1, v2, vm}, where each vi, i = 1, 2, ( ( ( ( m, is a linear combination of the vectors in S. Prove that w = b1v1 + b2v2 + ( ( ( + bm vm is a linear combination of the vectors in S.
Experiment in your software with the use of reduced row echelon form for the vectors in R2 given here. Are they linearly independent or linearly dependent? Compare the theoretical answer with the computational answer from your software.(a)(b) (c)
Find a basis for the subspace W of M33 consisting of all symmetric matrices.
(a) All vectors of the formWhere ( = 0 (b) All vectors of the form (c) All vectors of the form Where ( - b + 5c = 0
Find a basis for R3 that includes(a) The vector(b) The vectors And
Find a basis for M23. What is the dimension of M23? Generalize to Mmn.
Prove that if {v1, v2, ( ( ( ( vk} is a basis for a vector space V, then {cv1, v2, ( ( ( ( vk}, for c ( 0, is also a basis for V.
Prove that the vector space P of all polynomials is not finite-dimensional. Suppose that {p1(t), p2(t), ( ( ( ( Pk(t)} is a finite basis for P. Let dj = degree Pj (t). Establish a contradiction.
Show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W ( dim V.
Show that if W is a subspace of a finite-dimensional vector space V and dim W = dim V, then W = V.
Prove that the subspaces of R3 are {0}, R3 itself, and any line or plane passing through the origin.
Let S = (v1, v2, ( ( ( ( vn} be a set of nonzero vectors in a vector space V such that every vector in V can be written in one and only one way as a linear combination of the vectors in S. Prove that S is a basis for V.
Suppose that {v1, v2, ( ( ( ( vn} is a basis for Rn. Show that if A is an n ( n nonsingular matrix, then{Av1, Av2, ( ( ( ( Avn}is also a basis for Rn.
Suppose that {v1, v2, ( ( ( ( vn} is a linearly independent set of vectors in Rn and let A be a singular matrix. Prove or disprove that {Av1, Av2, ( ( ( ( Avn} is linearly independent.
Show that the set of matricesforms a basis for the vector space M22.
Let S = {x1, x2, ( ( ( ( xk} be a set of solutions to a homogeneous system Ax = 0. Show that every vector in span S is a solution to Ax = 0.
(a) Show that the zero matrix is the only 3 ( 3 matrix whose null space has dimension 3. (b) Let A be a nonzero 3 ( 3 matrix and suppose that Ax = 0 has a nontrivial solution. Show that the dimension of the null space of A is either 1 or 2.
Matrices A and B are m ( n, and their reduced row echelon forms are the same. What is the relationship between the null space of A and the null space of B?
In Example 3 we showed that an appropriate choice of basis could greatly simplify the computation of the values of a sequence of the form Av, A2v, A3v, ( ( ( ( Exercises 1 and 2 require an approach similar to that in Example 31. LetAnd(a) Show that S is a basis for R2.(b) Find [v] S.(c) Determine a
Let L: V ( W be an isomorphism of vector space V onto vector space W. (a) Prove that L(0V) =0W. (b) Show that L(v - w) = L(v) - L(w).
Find an isomorphism L: Rn ( Rn.
(a) Show that M22 is isomorphic to R4? (b) What is dim M22?
Let V be the subspace of the vector space of all real-valued continuous functions that has basis S = {et, e-t}. Show that V and R2 are isomorphic.
Let S = (v1, v2, ( ( (( vn) be an ordered basis for the n-dimensional vector space V, and let v and w be two vectors in V. Show that v = w if and only if [v]S = [w]S.
Show that if S is an ordered basis for an n-dimensional vector space V, v and w are vectors in V, and c is a scalar, thenAnd
Let MS be the n ( n matrix whose jth column is v and let MT be the n ( n matrix whose jth column is Wj. Prove that MS and MT are nonsingular. Consider the homogeneous systems MSx = 0 and MTx = 0.)Let S = [v1, v2, ( ( ( ( vn} and T = {w1, w2, ( ( ( ( wn} be ordered bases for the vector space Rn.
If v is a vector in V, show thatAndLet S = [v1, v2, ( ( ( ( vn} and T = {w1, w2, ( ( ( ( wn} be ordered bases for the vector space Rn.
(a) Use Equation (3) and Exercises 39 and 40 to show that(b) Show that PS(T is nonsingular.(c) Verify the result in part (a) of Example 4.Let S = [v1, v2, ( ( ( ( vn} and T = {w1, w2, ( ( ( ( wn} be ordered bases for the vector space Rn.
Let S be an ordered basis for n-dimensional vector space V. Show that if {w1, w2, ( ( ( ( wk} is a linearly independent set of vectors in V, thenIs a linearly independent set of vectors in Rn?
Let S = {v1, v2, ( v"} be an ordered basis for an n-dimensional vector space V. Show thatis an ordered basis for Rn.
(a) If A is a 3 ( 4 matrix, what is the largest possible value for rank A? (b) If A is a 4 ( 6 matrix, show that the columns of A are linearly dependent? (b) If A is a 5 ( 3 matrix, show that the rows of A are linearly dependent?
Let A be an n ( n matrix. Show that rank A = n if and only if the columns of A are linearly independent.
Let S = (v1, v2, ( ( ( ( vk} be a basis for a subspace V of Rn that is obtained by the method of Example 1. IfBelongs to V and the leading 1's in the reduced row echelon form from the method in Example 1 occur in columns j1, j2, ( ( ( ( jk then show thatv = (j1 v1, + (j2 v2 + ( ( ( + (jk vk.
Let A be an m ( n matrix. Show that the linear system Ax = b has a solution for every m ( 1 matrix b if and only if rank A = m.
Let A be an m ( n matrix with m ( n. Show that either the rows or the columns of A are linearly dependent.
Suppose that the linear system Ax = b, where A is m ( n, is consistent (i.e., has a solution). Prove that the solution is unique if and only if rank A = n.
Is it possible that all nontrivial solutions of a homogeneous system of 5 equations in 7 unknowns be multiples of each other? Explain.
Show that a set S = {v1, v2, ( ( ( ( vn} of vectors in Rn (Rn) spans Rn (Rn) if and only if the rank of the matrix whose jth column (jth row) is vj is n.
Show that in R3, (a) i ∙ i = j ∙ j = k ∙ k = 1; (b) i ∙ j = i ∙ k = j ∙ k = 0.
Which of the vectorsAre (a) Orthogonal?(b) In the same direction?(c) In opposite directions?
Suppose that an airplane is flying with an air speed of 260 kilometers per hour while a wind is blowing to the west at 100 kilometers per hour. Indicate on a figure the appropriate direction that the plane must follow to fly directly south. What will be the resultant speed?
Let θ be the angle between the nonzero vectors u and v in R2 or R3. Show that if u and v are parallel, then cos θ = ±1.
Show that the only vector x in R2 or R3 that is orthogonal to every other vector is the zero vector.
Let u be a fixed vector in R2 (R3). Prove that the set V of all vectors v in R2 (R3) such that u and v are orthogonal is a subspace of R2 (R3).
Let S = (v1, v2, v3) be a set of nonzero vectors in R3 such that any two vectors in 5 are orthogonal. Prove that S is linearly independent.
Prove that for any vectors u, v, and w in R2 or R3 and any scalar c, we have (a) (u + cv) ∙ w = u ∙ w + c(v ∙ w); (b) u ∙ (cv) = c(u ∙ v); (c) (u + v) ∙ (cw) = c(u ∙ w) + c(v ∙ w).
Prove that the angles at the base of an isosceles triangle are equal.
Prove that a parallelogram is a rhombus, a parallelogram with four equal sides, if and only if its diagonals are orthogonal.
Prove the Jacobi identity (u x v) x w + (v x w) x u + (w x u) x v = 0.
Find the point of intersection of the line of intersection of the given planes. (a) 2x + 3y - 4z + 5 = 0 and -3x + 2y + 5z + 6 = 0 (b) 3x - 2y - 5z + 4 = 0 and 2x + 3y + 4z + 8 = 0.
(a) Show that the graph of an equation of the form given in (10), with a, b, and c not all zero, is a plane with normal v = ai + bj + ck. (b) Show that the set of all points on the plane ax + by + cz = 0 is a subspace of R3. (c) Find a basis for the subspace given by the plane 2x - 3y + 4z = 0.
Let u = u1i + u2j + u3k, 3k, v = v1i + v2j + u3k, and w = w1i + w2j + w3k be vectors in P3. Show that
If (x1, y1) and (x2, y2) are distinct points in the plane, show thatIs the equation of the line through (x1, y1) and (x2, y2). Use this result to develop a test for collinearity of three points.
Let Pi(xi, yi, zi), i = 1, 2, 3, be three points in 3-space. Show thatis the equation of a plane (see Section 5.2) through points Pi, i = 1,2,3.
Prove properties (a) through (h) for the cross product operation.
Verify that each of the cross products u x v in Exercise 1 is orthogonal to both u and v.In Exercise 1(a) u = 2i + 3j + 4k, v = -i + 3j - k(b) u = i + k, v = 2i + 3j - k(b) u = i - j + 2k, v = 3i - 4j + k(d)
Show that u and v are parallel if and only if u x v = 0.
Prove the parallelogram law for any two vectors in an inner product space: |u + v||2 + ||u-v||2 = 2||u||2 + 2||v||2
Let V be an inner product space. Prove that if u and v are any vectors in V, then ||u + v||2 = ||u||2 + ||v||2 if and only if (u, v) = 0, that is, if and only if u and v are orthogonal. This result is known as the Pythagorean Theorem.
Let V be an inner product space. If u and v are vectors in V, Show that (u, v) = ¼ ||u + v||2 -1/4 ||u-v||2
Let V be an inner product space and u a fixed vector in V. Prove that the set of all vectors in V that are orthogonal to u is a subspace of V.
If V is an inner product space, prove that the distance function of Definition 5, 3 satisfies the following properties for all vectors u, v, and w in V: (a) d(u. v) ≥ 0 (b) d(u, v) = 0 if and only if u = v (c) d(u, v) = d(v, u) (d) d(u, v) ≤ d(u, w) + d(w, v)
Let C = [Cij] be an n x n positive definite symmetric matrix and let V be an n-dimensional vector space with ordered basis S = {u1, u2,... un}. For v = a1u1+ a2u2+...+anun and w = b1u1+b2u2+ ... + bnun in V defineProve that this defines an inner product on V.
In the Euclidean space Rn with the standard inner product, prove that (u, v) = uTv.
Consider Euclidean space R4 with the standard inner product and let(a) Prove that the set W consisting of all vectors in R4 that are orthogonal to both u1 and u2 is a subspace of R4. (b) Find a basis for W.
Let V be an inner product space. Show that if v is orthogonal to w1, w2....wk, then v is orthogonal to every vector in span {w1, w2..... wk}.
Suppose that {v1, v2.... vn} is an orthonormal set in Rn with the standard inner product. Let the matrix A be given by A = [v1 v2 ∙ ∙ ∙ vn]. Show that A is non-singular and compute its inverse. Give three different examples of such a matrix in R2 or R3.
If A is nonsingular, prove that ATA is positive definite.
If C is positive definite, and x ≠ 0 is such that Cx = kx for some scalar k, show that k > 0.
Let C be positive definite and r any scalar. Prove or disprove: rC is positive definite.
If B and C are n x n positive definite matrices, show that B + C is positive definite.
Let V be an inner product space. Show the following: (a) ||0||=0. (b) (u, 0) = (0, u) = 0 for any u in V. (c) If (u, v) = 0 for all v in V, then u = 0. (d) If (u, w) = (v, w) for all w in V, then u = v. (e) If (w, u) = (w, v) for all w in V, then u = v.

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