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mathematics
linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

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 = 8 -3 0 -3 0 -1 0 - 1 -1 8 8-3
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) If PB is the change-of-coordinates matrix, then [x]B = PB x, for x in V.
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
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
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 =
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
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
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.
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
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.
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
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) =
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.
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.
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
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.
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,
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
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
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
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.
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
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
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
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
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
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
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,
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) =
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
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
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

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