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linear algebra and its applications

Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions

Letand H = Span {V1, V2, V3). It can be verified that 4v1 +5v2 - 3v3 = 0. Use this information to find a basis for H. There is more than one answer. 4 V1 -------- = -3 V2 = 9 V3 = 11 -2 7 7 6
For the matrices (a) Find k such that Nul A is a subspace of Rk, (b) Find k such that Col A is a subspace of Rk. A = [1 -3 9 0 −5]
LetIt can be verified that v1 - 3v2 + 5v3 = 0. Use this information to find a basis for H = Span {v1, v2, v3}. V1 = 7 4 -9 -5 V₂ = 4 -7 2 5 V3 = -5 3 4
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) In some cases, a plane in R3 can be isomorphic to R2.
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) The vector spaces P3 and R3 are isomorphic.
LetDetermine if w is in Col A. Is w in Nul A? -6 12 4-[¹3] and w- [²] w= 6 A
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) The correspondence [x]B ↦ x is called the coordinate mapping.
Let Determine if w is in Col A. Is w in Nul A? A = -8 -2 4 0 864 -97 8 and w= 4 27 1 -2
LetSince the coordinate mapping determined by B is a linear transformation from R2 into R2, this mapping must be implemented by some 2 × 2 matrix A. Find it. B = -3 {-1] []} -2 7
Let B = {b1,...,bn} be a basis for a vector space V. Explain why the B-coordinate vectors of b1,...,bn are the columns e1,...,en of the n x n identity matrix.
Let B = {b1,...,bn} be a basis for Rn. Produce a description of an n x n matrix A that implements the coordinate mapping x ↦ [x]B. (See Exercise 25.)Data from in Exercise 25LetSince the coordinate mapping determined by B is a linear transformation from R2 into R2, this mapping must be implemented
A denotes an m x n matrix. Justify each answer. (T/F) The null space of A is the solution set of the equation Ax = 0.
Justify each answer.(T/F) A single vector by itself is linearly dependent.
Justify each answer.(T/F) A linearly independent set in a subspace H is a basis for H.
Justify each answer.(T/F) If H = Span {b1,...,bp), then {b1,...,bp} is a basis for H.
Justify each answer(T/F) If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V.
Let S be a finite set in a vector space V with the property that every x in V has a unique representation as a linear combination of elements of S. Show that S is a basis of V.
Justify each answer.(T/F) If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.
Justify each answer(T/F) A vector is any element of a vector space.
Concern a vector space V, a basis B = {b1,...,bn), and the coordinate mapping x ↦ [x] B.Show that a subset {u1,...,up} in V is linearly independent if and only if the set of coordinate vectors {[u1]B,...,[Up]B} is linearly independent in Rn. + Cpup = 0 The zero vector in V [C₁u₁ ++Cpup lg =
Suppose {v1,...,v4} is a linearly dependent spanning set for a vector space V. Show that each w in V can be expressed in more than one way as a linear combination of v1,...,v4.
Justify each answer(T/F) An arrow in three-dimensional space can be considered to be a vector.
Justify each answer.(T/F) A basis is a linearly independent set that is as large as possible.
Justify each answer(T/F) If u is a vector in a vector space V, then (-1) u is the same as the negative of u.
Concern a vector space V, a basis B = {b1,...,bn), and the coordinate mapping x ↦ [x] B.Show that the coordinate mapping is one-to-one. Suppose [u]B = [w]B for some u and w in V, and show that u = w.]
A denotes an m x n matrix. Justify each answer.(T/F) A null space is a vector space.
Use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work. {1+ 2t³, 2+1-3t², -t +2t²-t³}
Justify each answer.(T/F) A basis is a spanning set that is as large as possible.
Justify each answer(T/F) A subset H of a vector space V is a subspace of V if the zero vector is in H.
A denotes an m x n matrix. Justify each answer.(T/F) The null space of an m x n matrix is in Rm.
Use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work. {1-2t²t³,t + 2t³, 1+t2t²}
Justify each answer.(T/F) The standard method for producing a spanning set for Nul A, described, sometimes fails to produce a basis for Nul A.
Justify each answer(T/F) A vector space is also a subspace.
Concern a vector space V, a basis B = {b1,...,bn), and the coordinate mapping x ↦ [x] B.Show that the coordinate mapping is onto Rn. That is, given any y in Rn, with entries y1,...,yn, produce u in V such that [u]B = y.
Justify each answer.(T/F) In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.
Justify each answer(T/F) A subspace is also a vector space.
A denotes an m x n matrix. Justify each answer.(T/F) The column space of A is the range of the mapping x ↦ Ax.
Justify each answer.(T/F) If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
Use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work. {(1-t)², t2t² +t³, (1 - 1)³}
Justify each answer(T/F) R2 is a subspace of R3.
Justify each answer.(T/F) If A and B are row equivalent, then their row spaces are the same.
A denotes an m x n matrix. Justify each answer.(T/F) Col A is the set of all solutions of Ax = b.
Concern a vector space V, a basis B = {b1,...,bn), and the coordinate mapping x ↦ [x] B.Given vectors u1,...,up, and w in V, show that w is a linear combination of u1,...,up if and only if [w]B is a linear combination of the coordinate vectors [u1]B,...,[up]B.
Justify each answer.(T/F) Row operations preserve the linear dependence relations among the rows of A.
Justify each answer(T/F) The polynomials of degree two or less are a subspace of the polynomials of degree three or less.
A denotes an m x n matrix. Justify each answer.(T/F) If the equation Ax = b is consistent, then Col A = Rm.
Show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and -u + u = 0 for all u.Complete
Suppose R4 = Span {v1,...,v4). Explain why {v1,...,v4} is a basis for R4.
Justify each answer(T/F) A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) The zero vector of V is in H(ii) u, v, and u + v are in H(iii) c is a scalar and cu is in H
A denotes an m x n matrix. Justify each answer.(T/F) Nul A is the kernel of the mapping x ↦ Ax.
Show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and -u + u = 0 for all u.Complete
A denotes an m x n matrix. Justify each answer.(T/F) The kernel of a linear transformation is a vector space.
Let B = {v1,...,Vn} be a linearly independent set in Rn. Explain why B must be a basis for Rn.
A denotes an m x n matrix. Justify each answer.(T/F) The range of a linear transformation is a vector space.
Use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work. {(2-1)³, (3-1)², 1+ 6t - 5t² +1³}
Letand let H be the set of vectors in R3 whose second and third entries are equal. Then every vector in H has a unique expansion as a linear combination of v1, v2, v3, because for any s and t . Is {v1, v2, v3} a basis forH? Why or why not? ------- V1 = 0 = 1 V3 = 1 0
Use coordinate vectors to test whether the following sets of polynomials span P2. Justify your conclusions.a.b. {1 - 3t + 5t², -3 + 5t - 7t²,-4+ 5t - 6t², 1-t²}
Leta. Use coordinate vectors to show that these polynomials form a basis for P2.b. Consider the basis B = {P1, P2, P3} for P2. Find q in P2, given that P₁(t) = 1+t², P₂ (t) = t - 3t², P3 (t) = 1 + 1-31². 2
Show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and - u + u = 0 for all u.Fill in
Show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and - u + u = 0 for all u.Fill in
A denotes an m x n matrix. Justify each answer.(T/F) Col A is the set of all vectors that can be written as Ax for some x.
In the vector space of all real-valued functions, find a basis for the subspace spanned by {sin t, sin 2t, sin t cos t}.
A denotes an m x n matrix. Justify each answer.(T/F) The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
Let V be the vector space of functions that describe the vibration of a mass–spring system. (Refer to Exercise 19) Find a basis for V .Data from in Exercise 19If a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass–spring system will begin to
A denotes an m x n matrix. Justify each answer.(T/F) The row space of A is the same as the column space of AT.
Determine whether the sets of polynomials form a basis for P3. Justify your conclusions. 3+7t,5+t2t³, t2t², 1+ 16t - 6t² + 2t³
It can be shown that a solution of the system below is x1 = 3, x2 = 2, and x3 = -1. Use this fact and the theory from this section to explain why another solution is x1 = 30, x2 = 20, and x3 = -10. X1 - 3х2 - 3x3 = 0 -2x1 + 4x2 + 2x3 = 0 -xi + 5x2 +7x3 = 0
Show that every basis for Rn must contain exactly n vectors.Let S = {v1,...,vk} be a set of k vectors in Rn, with k n.Data from in Theorem If a set S = {₁,..., Vps in R" contains the zero vector, then the set is linearly dependent. PROOF By renumbering the vectors, we may suppose v₁ = 0. Then
Concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying figure. The vectorsin R3 form a basis for the unit cell shown on the right. The numbers here are Angstrom units (1 A = 10-8 cm). In alloys of titanium, some additional atoms may be in
Let H = Span {v1, v2} and B = {v1, v2}. Show that x is in H and find the B-coordinate vector of x, for V₁ = 11 -5 10 7 V2 14 -8 13 10 X = 19 -13 18 15
Concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying figure. The vectorsin R3 form a basis for the unit cell shown on the right. The numbers here are Angstrom units (1 A = 10-8 cm). In alloys of titanium, some additional atoms may be in
Determine whether the sets of polynomials form a basis for P3. Justify your conclusions. 5-3t+41² +21³,9 +1 +8t² - 6t³, 6-2t + 5t², t³
Let H = Span {v1, v2, v3) and B = {v1, v2, v3}. Show that B is a basis for H and x is in H, and find the B-coordinate vector of x, for || -6 40 4 -9 4 V2 || 8 -3 7 V3 = -9 5 5 ∞0 m -8 3 || 4 7 -8 3
Show that every basis for Rn must contain exactly n vectors.Let S = {v1,...,vk} be a set of k vectors in Rn, with k > n. Use a theorem from Chapter 1 to explain why S cannot be a basis for Rn.Data from in Theorem 4 If a set S = {₁,..., Vps in R" contains the zero vector, then the set is
Define T : P2 → R2 by T For instance, if p(t) = 3 + 5t + 7t2, then Ta. Show that T is a linear transformation.b. Find a polynomial p in P2 that spans the kernel of T, and describe the range of T. (P) = P(0) P(1)
Consider the following two systems of equations:It can be shown that the first system has a solution. Use this fact and the theory from this section to explain why the second system must also have a solution. 5x1 + x2 3x3 = 0 -9x1 + 2x₂ + 5x3 = 1 4x1 + x₂ - 6x3 = 9 - 5x1 + x2 3x3 = 0 -9x1
Repeat Exercise 45 for the functions f(t) = 3 sin t - 4 sin3 t, g(t) = 1 - 8 sin2 t + 8 sin4 t, h(t) = 5 sin t - 20 sin3 t + 16 sin5 t, in the vector space Span {I, sin t, sin2 t, ..., sin5 t}.Data from in Exercise 45The vector space H = Span {1, cos2 t, cos4 t, cos6 t} contains at least two
A denotes an m x n matrix. Justify each answer.(T/F) The null space of A is the same as the row space of AT.
Let H = Span {u1, u2, u3} andK = Span {v1, v2, v3}, whereFind bases for H, K, and H + K. U₁ || V1 = 3 0 -4 3 2 1] U₂ = V₂ = 3 1 1 9 -4 1 U13 = V3 = 2 -3 6 -5 7 6 5
Show that {1, cos t, cos2 t,...,cos6 t} is a linearly independent set of functions defined on R. Use the method of Exercise 47.Data from in Exercise 47Show that {t, sin t, cos 2t, sin t cos t} is a linearly independent set of functions defined on R. Start by assuming thatEquation (5) must hold for
Determine whether w is in the column space of A, the null space of A, or both, where W= 1 -3 A = 7 -5 6-4 1 0-2 7 -3 7 1 -1 9 -11 -9 19
Reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T: V → W be a linear transformation, and let {v1,...,vp} be a subset of V.Show that if {v1,...,vp} is linearly
Show that {t, sin t, cos 2t, sin t cos t} is a linearly independent set of functions defined on R. Start by assuming thatEquation (5) must hold for all real t, so choose several specific values of t (say, t = 0, .1, .2) until you get a system of enough equations to determine that all the cj must be
Reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T: V → W be a linear transformation, and let {v1,...,vp} be a subset of V.Suppose that T is a one-to-one
Consider the polynomials p1 (t) = 1 + t2 and p2(t) = 1- t2. Is {P1, P2} a linearly independent set in P3? Why or why not?
Consider the polynomials p1(t) = 1 + t, p2(t) = 1 - t, and P3 (t) = 2 (for all t). By inspection, write a linear dependence relation among P1, P2, and p3. Then find a basis for Span (P1, P2, P3}.
Let V be a vector space that contains a linearly independent set {u1, u2, u3, u4}. Describe how to construct a set of vectors {v1, v2, v3, v4} in V such that {v1, v3} is a basis for Span {v1, v2, v3, v4}.
Define T: C[0, 1] → C[0, 1] as follows: For f in C[0, 1], let T(f) be the antiderivative F of f such that F(0) = 0. Show that T is a linear transformation, and describe the kernel of T.
Determine whether w is in the column space of A, the null space of A, or both, where W= 2 1 0 A = -8 -5 10 -8 3 5282 0 1-2 6 -3 0 5-2 -2 1
Let a1,....,a5 denote the columns of the matrix A, wherea. Explain why a3 and a5 are in the column space of B. b. Find a set of vectors that spans Nul A. c. Let T: R5 → R4 be defined by T(x) = Ax. Explain why T is neither one-to-one nor onto. A = 53∞ 2 5 8 2 122 0 3 2 -1 -12 12 -2 34 224 4 4
Let H = Span {V1, V2} and K = Span {V3, V4}, whereThen H and K are subspaces of R3. In fact, H and K are planes in R3 through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. w can be written as c1v1 + c2v2 and also as c3V3 + C4V4. To build w,
Justify each answer.(T/F) If A and B are n x n, then (A + B) (A - B) = A2 - B2.
Display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. A = ~ 1 1 -2 4 1 0 0 -2 -1 -2 1 0 0 0 1 -6 9 9 5 6 9 5 0 1 0 -3 0 0 บน 5 1 1 4 -3 -2 -9 4 -7 -2 0
Assume that the matrices mentioned below have appropriate sizes. Justify each answer.(T/F) Every square matrix is a product of elementary matrices.
Let H = Span{V1, V2} and B = {V1, V2}. Show that x is in H, and find the B-coordinate vector of x, when V₁ = 12 -4 9 5 V2 = 15 -7 12 8 X = 19 -11 16 12
Let H = Span {V1, V2, V3} and B = {V1, V2, V3}. Show that B is a basis for H and x is in H, and find the B-coordinate vector of x, when V₁ = -5 4 -3 2 V₂ = 7 -5 3 -3 V3 = -8 6 -4 3 X = 8 -9
Construct a nonzero 3 x 3 matrix A and a vector b such that b is not in Col A.
Use Exercise 35 to show that if A and B are bases for a subspace W of R", then A cannot contain more vectors than B, and, conversely, B cannot contain more vectors than A.Data from in Exercise 35Suppose vectors b1,...,bp span a subspace W, and let {a1,...,aq} be any set in W containing more than p
Construct a nonzero 3 x 3 matrix A and a nonzero vector b such that b is in Nul A.
What can you say about the shape of an m x n matrix A when the columns of A form a basis for Rm?

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