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linear algebra

Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

Find the coordinate vector [v]B ofWith respect to the orthogonal basis
The coordinate vector of a vector v with respect to an orthonormal basisIfFind all possible vectors v.
If Q is an orthogonal n ( n matrix and {v1( ( ( ( ( vk} is an orthonormal set in, prove that {Q v1( ( ( ( ( Q vk} is an orthonormal set.
If Q is an n ( n matrix such that the angles ˆ (Qx, Qy) and ˆ (x, y) are equal for all vectors x and y in , prove that Q is an orthogonal matrix.
W is the line in with general equation 2x - 5y = 0
Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, list all of the axioms that fail to hold.The set of all vectors in R2 of the formwith the usual vector addition and scalar multiplication
Finish verifying that P2 is a vector space (see Example 6.3). Example 6.3 Let P2 denote the set of all polynomials of degree 2 or less with real coefficients. Define addition and scalar multiplication in the usual way. (See Appendix D.) If p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 are in
Finish verifying that P is a vector space (see Example 6.4).In example(a)(b) (c)
In Exercises 1-3, determine whether the given set, together with the specified operations of addition and scalar multiplication, is a complex vector space. If it is not, list all of the axioms that fail to hold.1. The set of all vectors in C2 of the formWith the usual vector addition and scalar
In Exercises 1-3, determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space over the indicated Zp, If it is not, list all of the axioms that fail to hold.1. The set of all vectors in Zn2 with an even number of l s, over Z2 with
In Exercises 1-2, use Theorem 6.2 to determine whether W is a subspace of V1.2. 3.
Let V be a vector space with subspaces U and W. Prove multiplication that U ∩ W is a subspace of V.
Let V be a vector space with subspaces U and W. Give an example with v = [R2 to show that U U W need not be a subspace of V.
Let V be a vector space with subspaces U and W. Define the s u m of U a n d W to be U + W = {u + w: u is in U, w is in W} (a) If V = R3, U is the x-axis, and W is the y-axis, what is U + W? (b) If U and W are subspaces o f a vector space V, prove that U + W is a subspace of V.
If U and V are vector spaces, define the Cartesian product of U and V to be U X V = {(u, v): u is in U and v is in V} Prove that U × V is a vector space.
Let W be a subspace of a vector space V. Prove that Δ = {(w, w): w is in W} is a subspace of V × V.
In Exercise 1 and 2, letAnd Determine whether C is in span(A, B). 1. 2.
In Exercises 1 and 2, let p(x) = 1 - 2x, q(x) = x - x2, and r(x) = - 2 + 3x + x2. Determine whether s(x) is in span(p(x), q(x), r(x)). 1. s(x) = 3 - 5x - x2 2. s(x) = 1 + x + x2
In Exercises 1-2, let f(x) = sin2x and g(x) = cos2x. Determine whether h(x) is in span(f(x), g(x)). 1. f(x) = 1 2. h(x) = cos 2x 3. h(x) = sin 2x
Is P2 spanned by 1 + x, x + x2, 1 + x2?
Is P2 spanned by 1 + x + 2x2, 2 + x + 2x2, - 1 + x + 2x2?
Prove that every vector space has a unique zero vector.
Prove that for every vector v in a vector space V, there is a unique v' in V such that v + v' = 0.
In Exercises 1-3, test the sets of matrices for linear independence in M22. For those that are linearly dependent, express one of the matrices as a linear combination of the others.1.2. 3.
In Exercises 1-2, test the sets of functions for linear independence in P, For those that are linearly dependent, express one of the functions as a linear combination of the others. 1. {1, sin x, cos x} 2. {1, sin2 x, cos2 x} 3. {ex, e-x}
If f and g are in b(j), the vector space of all functions with continuous derivatives, then the determinantis called the Wronskian off and g [named after the Polish-French mathematician Josef Maria HoeneWronski (1 776- 1 853), who worked on the theory of determinants and the philosophy of
In general, the Wronskian of f1, ˆ™ ˆ™ ˆ™ fn in b(n - 1) is the Determinantand f1, . . . ,fn are linearly independent, provided W(x) is not identically zero. Repeat Exercises 10-14 using the Wronskian test.
Let {u, v, w} be a linearly independent set of vectors in a vector space V. (a) Is {u + v, v + w, u + w} linearly independent? Either prove that it is or give a counterexample to show that it is not. (b) Is {u - v, v - w, u - w} linearly independent? Either prove that it is or give a counterexample
In Exercises 1-3, determine whether the set B is a basis for the vector space V1.2.3. V = M22,
Find the coordinate vector ofwith respect to the basis B = {E22, E21, E12, E11} of M22.
Find the coordinate vector ofwith respect to the basis of M22.
Find the coordinate vector of p (x) = 1 + 2x + 3x2 with respect to the basis B = {l + x, 1 - x, x2} of P2.
Find the coordinate vector of p (x) = 2 - x + 3x2 with respect to the basis B = {l, 1 + x, - 1 + x2} of P2.
Let B be a set of vectors in a vector space V with the property that every vector in V can be written uniquely as a linear combination of the vectors in B. Prove that B is a basis for V.
Finish the proof of Theorem 6.7 by showing that if {[u1] 8, . . . , [uk]B} is linearly independent in Rn then {u1, . . . , uk} is linearly independent in V.
Let {u1, . . . , um} be a set of vectors in an n-dimensional vector space V and let B be a basis for V. Let S = {[u1]B , . . . , [um] B} be the set of coordinate vectors of {u1, . . . , um} with respect to B. Prove that span(u1, . . . , um) = V if and only if span(S) = Rn.
In Exercises 1-3, find the dimension of the vector space V and give a basis for V. 1. V = {P(x) in P2: p(0) = 0} 2. V = {p(x) in P2: p(1) = 0} 3. V = {p(x) in P2: xp'(x) = p(x)}
Find a formula for the dimension of the vector space of symmetric n × n matrices.
Find a formula for the dimension of the vector space of skew-symmetric n × n matrices.
Let U and W be subspaces of a finite-dimensional vector space V. Prove Grassmann's Identity:dim ( U + W) = dim U + dim W - dim( U ∩ W)
Let U and V be finite-dimensional vector spaces. (a) Find a formula for dim( U × V) in terms of dim U and dim V. (See Exercise 49 in Section 6. 1.) (b) If W is a subspace of V, show that dim Δ = dim W, where Δ = {(w, w): w is in W}.
Prove that the vector space P is infinite-dimensional.
Extend {l + x, 1 + x + x2} to a basis for P2.
Find a basis for span (l, 1 + x, 2x) in P1.
In Exercises 1-3, test the sets of polynomials for linear independence. For those that are linearly dependent, express one of the polynomials as a linear combination of the others.1. {x, 1 + x} in P12. {1 + x, 1 + x2, 1 - x + x2} in P2
Find a basis for span (l - 2x, 2x - x2, 1 - x2, 1 + x2) in P2.
Find a basis for span (l - x, x - x2, 1 - x2, 1 - 2x + x2) in P2.
Find a basis for span(sin2x, cos2x, cos 2x) in P.
Let S = {v1, . . . , vn} be a linearly independent set in a vector space V. Show that if v is a vector in V that is not in span(S), then S' = {v1, . . . , vn, v} is still linearly independent.
Let S = {v1, . . . , vn} be a spanning set for a vector space V. Show that if vn is in span (v1, . . . , vn-1), then S' = {v1, . . . , vn-1} is still a spanning set for V.
Let {v1, . . . , vn} be a basis for a vector space V and let c1, . . . , cn be nonzero scalars. Prove that {c1v1, . . . , cnvn} is also a basis for V.
Let {v1, . . . , vn}be a basis for a vector space V. Prove that {v1, v1 + v2, v1 + v2 + v3 , . . . , v1 + . . . + vn} is also a basis for V.
Let a0, a1, . . . , an be n + 1 distinct real numbers. Define polynomials p0(x), p1 (x), . . . , Pn(x) byThese are called the Lagrange polynomials associated with a0, a1, . . . , an [Joseph-Louis Lagrange (1736- 1813) was born in Italy but spent most of his life in Germany and France. He made
(a) Prove that the set B = {p0 (x), p1 (x) , . . . , Pn(x)} of Lagrange polynomials is linearly independent in Pn. (b) Deduce that B is a basis for Pn.
If q(x) is an arbitrary polynomial in Pn' it follows from Exercise 60(b) that q (x) = c0p0(x) + . . . + cnpn(x).....................(1) for some scalars c0, . . . , cn. (a) Show that ci = q (ai) for i = 0, . . . , n, and deduce that q(x) = q(a0)p0(x) + · · · + q(an)Pn(x) is the unique
Use the Lagrange interpolation formula to show that if a polynomial in Pn has n + 1 zeros, then it must be the zero polynomial.
Find a formula for the number of invertible matrices in Mnn (Zp).
In Exercises 1-2:(a) Find the coordinate vectors [x]B and [x]C of x with respect to the bases B and C, respectively.(b) Find the change-of-basis matrix PC†B from B to C.(c) Use your answer to part (b) to compute [x]C, and compare your answer with the one found in part (a).(d) Find the
In Exercises 1 and 2, follow the instructions for Exercises 1-4 using f(x) instead of x. 1. f(x) = 2 sin x - 3 cos x, B = {sin x + cos x, cos x}, C = {sin x + cos x, sin x - cos x} in span(sin x, cos x) 2. f(x) = sin x, B = {sin x + cos x, cos x}, C = {cos x - sin x, sin x + cos x} in span(sin x,
Rotate the xy-axes in the plane counterclockwise through an angle θ = 60° to obtain new x' y' -axes. Use the methods of this section to find (a) the x' y'-coordinates of the point whose xy-coordinates are (3, 2) and (b) the xy-coordinates of the point whose x'y' -coordinates are (4, - 4).
Repeat Exercise 13 with θ = 135°. In exercise Rotate the xy-axes in the plane counterclockwise through an angle θ = 60° to obtain new x' y' -axes. Use the methods of this section to find (a) the x' y'-coordinates of the point whose xy-coordinates are (3, 2) and (b) the xy-coordinates of the
Let B and C be bases for R2. IfAnd the change-of-basis matrix from B to C is Find B.
Let B and C be bases for P2. If B = {x, 1 + x, 1 -x + x2} and the change-of-basis matrix from B to C isFind C.
Express p (x) = 1 + 2x - 5x2 as a Taylor polynomial about a = 1. In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form p(x) = a0 + a1(x - a) + a2(x - a)2 + · · · + an(x - a)n where an ≠ 0. In other words, it is a polynomial that has been expanded in
Express p (x) = 1 + 2x - 5x2 as a Taylor polynomial about a = - 2. In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form p(x) = a0 + a1(x - a) + a2(x - a)2 + · · · + an(x - a)n where an ≠ 0. In other words, it is a polynomial that has been expanded in
Express p (x) = x3 as a Taylor polynomial about a = - 1. In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form p(x) = a0 + a1(x - a) + a2(x - a)2 + · · · + an(x - a)n where an ≠ 0. In other words, it is a polynomial that has been expanded in terms of
Express p (x) = x3 as a Taylor polynomial about a = t. In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form p(x) = a0 + a1(x - a) + a2(x - a)2 + · · · + an(x - a)n where an ≠ 0. In other words, it is a polynomial that has been expanded in terms of
Let B, C, and V be bases for a finite-dimensional vector space V. Prove that PD←C PC←B = PD←B
Let V be an n-dimensional vector space with basis B = {v1 , . . . , vn}. Let P be an invertible n X n matrix and set ui = p1iv1 + ∙ ∙ ∙ + pni vN for i = 1 , . . . , n. Prove that C = {u1 , . . . , un} is a basis for V and show that P = PB←C·
In Exercises 1-2, follow the instructions for Exercises 1-4 using p(x) instead of x. 1. p(x) = 2 - x, B = {1, x}, C = {x, 1 + x} in P1 2. p(x) = 1 + 3x, B = {1 + x, 1 - x}, C = {2x, 4} in P1
In Exercises 1 and 2, follow the instructions for Exercises 1-4 using A instead of x.1.2.
In Exercises 1-2, determine whether T is a linear transformation.1. T: M22 †’ M22 defined by2. T: M22 †’ M22 defined by 3. T: Mnn †’ Mnn defined by T(A) = AB, where B is a fixed n Ã— n matrix.
Show that the transformations S and T in Example 6.56 are both linear.
Let T: R2 †’ R3 be a linear transformation for whichFind
Let T: R2 †’ P2 be a linear transformation for whichFind
Let T: P2 → P2 be a linear transformation for which T(l) = 3 - 2x, T(x) = 4x - x2, and T(x2) = 2 + 2x2 Find T(6 + x - 4x2) and T(a + bx + cx2).
Let T: P2 → P2 be a linear transformation for which T(l + x) = 1 + x2, T(x + x2) = x - x2, T(1 + x2) = 1 + x + x2 Find T(4 - x + 3x2) and T(a + bx + cx2).
Let T: M22 †’ R be a linear transformation for whichFind
Let T: M22 †’ R be a linear transformation. Show that there are scalars a, b, c, and d such thatFor allIn M22.
Show that there is no linear transformation T: R3 †’ P2 such that
Let {v1, . . . , vn} be a basis for a vector space V and let T: V → V be a linear transformation. Prove that if T (v1) = V1, T (v2) = V2 . . . , T(vn) = vn, then T is the identity transformation on V.
Let T: Pn → Pn be a linear transformation such that T(xk) = kxk-1 for k = 0, 1 , . . . , n. Show that T must be the differential operator D.
Let v1, . . . , vn be vectors in a vector space V and let T: V → W be a linear transformation. (a) If {T(v1) , . . . , T(vn)} is linearly independent in W, show that {v1, . . . , vn} is linearly independent in V. (b) Show that the converse of part (a) is false. That is, it is not necessarily true
Define linear transformations S: R2 †’ M22 and T: R2 †’ R2 byCompute Can you compute If so, compute it.
Define linear transformations S: P1 → P2 and T: P2 → P1 by S(a + bx) = a + (a + b )x + 2bx2 and T(a + bx + cx2) = b + 2cx Compute (S ͦ T) (3 + 2x - x2) and (S ͦ T) (a + bx + cx2). Can you compute (T ͦ S) (a + bx)? If so, compute it.
Define linear transformations S: Pn → Pn and T: Pn → Pn by S(p (x)) = p(x + 1) and T(p(x)) = p'(x) Find (S ͦ T) (p (x)) and (T ͦ S) (p (x)).
Define linear transformations S: Pn → Pn and T: Pn → Pn by S(p (x)) = p (x + 1) and T(p(x)) = xp'(x) Find (S ͦ T) (p (x)) and (T ͦ S) (p (x)).
In Exercises 29 and 30, verify that S and T are inverses.1. S: R2 †’ R2 defined byand T: R2 †’ R2 defined by2. S: P1 †’ P1 defined by S(a + bx) = (- 4a + b) + 2ax and T: P1 †’ P1 defined by T(a + bx) = b/2 + (a + 2b)x
Let T: V → V be a linear transformation such that T o T = I. (a) Show that {v, T(v)} is linearly dependent if and only if T(v) = ± v. (b) Give an example of such a linear transformation with V = R2.
Let T: V → V be a linear transformation such that T ͦ T = T. (a) Show that {v, T(v)} is linearly dependent if and only if T(v) = v or T(v) = 0. (b) Give an example of such a linear transformation with V = R2.
Prove that S + T and cT are linear transformations. The set of all linear transformations from a vector space V to a vector space W is denoted by £(V, W) . If S and T are in £ (V, W), we can define the sum S + T of S and T by (S + T) (v) = S(v) + T(v) for all v in V If c is a scalar, we define
Prove that £ (V, W) is a vector space with this addition and scalar multiplication.
Let R, S, and T be linear transformations such that the following operations make sense. Prove that. (a) R ͦ (S + T) = R ͦ S + R ͦ T (b) c(R ͦ S) = (cR) ͦ S = R ͦ (cS) for any scalar c
Let T: M22 †’ M22 be the linear transformation defined by(a) Which, if any, of the following matrices are in ker( T)? (i) (ii) (iii) (b) Which, if any, of the matrices in part (a) are in range(T)?
In Exercises 1-3, determine whether the linear transformation T is (a) one-to-one and (b) onto.1. T: R2 †’ R2 defined by2. T: R2 †’ P2 defined by
Let T: M22 †’ R be the linear transformation defined by T(A) = tr(A).(a) Which, if any, of the following matrices are in ker(T)?(i)(ii) (iii) (b) Which, if any, of the following scalars are in range(T)? (i) 0 (ii) 2 (iii) ˆš2 (c) Describe ker(T) and range(T).
In Exercises 1-3, determine whether V and W are isomorphic. If they are, give an explicit isomorphism T: V → W. 1. V = D3 (diagonal 3 × 3 matrices), W = R3 2. V = S3 (symmetric 3 × 3 matrices), W = U3 (upper triangular 3 × 3 matrices) 3. V = S3 (symmetric 3 × 3 matrices), W = S'3
Show that T: Pn → Pn defined by T(p(x) ) = p(x) + p'(x) is an isomorphism.
Show that T: Pn → Pn defined by T(p(x) ) = p(x - 2) is an isomorphism.
Show that T: Pn → Pn defined by T(p(x)) = xnp(1/x) 1s an isomorphism.

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