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

Find an orthonormal basis for the subspace of R4 consisting of all vectors of the form
Find an orthonormal basis for the subspace of R4 consisting of all vectors [a b c d] such that a - b - 2c + d = 0.
Use the Gram-Schmidt process to transform the basisFor the subspace W of Euclidean space R3 into (a) An orthogonal basis; (b) An orthonormal basis.
Let S = {[l -1 0], [l 0 -l]| be a basis for a subspace W of the Euclidean space R3. (a) Use the Gram-Schmidt process to obtain an orthonormal basis for W. (b) Using Theorem 5.5, write u = [5 -2 -3] as a linear combination of the vectors obtained in part (a).
Prove that if T is an orthonormal basis for a Euclidean space and
Let W be the subspace of the Euclidean space R3 with basisLet (a) Find the length of v directly (b) Using the Gram-Schmidt process, transform S into an orthonormal basis T for W.
Let V be the Euclidean space of all 2 x 2 matrices with inner product defined by (A, B) = Tr(BT A).(a) Prove thatIs an orthonormal basis for V. (b) Use Theorem 5.5 to find the coordinate vector of With respect to S
LetBe a basis for a subspace W of the Euclidean space defined in Exercise 25. Use the Gram-Schmidt process to find an orthonormal basis for W.
Let u1, u2, ..., un be vectors in Rn. Show that if u is orthogonal to u1, u2,..., un then u is orthogonal to every vector in span {u1, u2,..., un}.
Let u be a fixed vector in Rn. Prove that the set of all vectors in Rn that are orthogonal to u is a subspace of Rn.
Let S = {u1, u2,..., uk} be an orthonormal basis for a subspace W of Euclidean space V that has dimension n > k. Discuss how to construct an orthonormal basis for V that includes S.
Let S = {v1, v2,..., vk} be an orthonormal basis for the Euclidean space V and {a1, a2,..., ak) be any set of scalars none of which is zero. Prove that T = {a1v1,a2v2, ...,akvk} is an orthogonal basis for V. How should the scalars a1, a2,..., ak be chosen so that T is an orthonormal basis for V?
In the proof of Theorem 5.8, show that the diagonal entries r" are nonzero by first expressing u, as a linear combination of v1, v2,..., vi, and then computing rii = (ui, wi).
Let W be the plane R3 given by the equationIn Exercise, find projWv for the given vector v and subspace W.(a)(b) (c)
Let(a) Find a basis for WŠ¥. (b) Describe WŠ¥ geometrically. (You may use a verbal or pictorial description.)
Let V be an inner product space show that orthogonal complement of V is the zero subspace is V itself
Show that if W is a subspace of an inner product space V that is spanned by a set of vectors S, then a vector u in V belongs to W⊥ if and only if u is orthougonal to every vector in S.
Let A be an m x n matrix. Show that every vector v in Rn can be written uniquely as w + u where w is in the null space of A and u is in the column space of AT.
Let W be a subspace of an inner product space V and let {w1, w2, ... wm} be an orthogonal basis for W. Show that if v is any vector in V, then
Let A be n x n and nonsingular. From the normal system of equations in (1), show that the least squares solution to Ax = b is x = A-1b.
The following data showing U.S. per capita health care expenditures (in dollars) is available from the National Center for Health Statistics (www/cdc.gov/nchs/ hus.htm) in the 2002 Chartbook on Trends in the Health of Americans. Year Per Capita Expenditures (in
The following data show the size of the U.S. debt per capita (in dollars). This information was constructed from federal government data on public debt (www.publicdebt.treas.gov/opd/opd.htm#history) and (estimated) population data (www.censi.is. gov.popest/archives/). (a) Determine the line of best
Solve Exercise 2 using QR-factorization of AIn Exercise 2
In Exercise 6, the least squares line to the data set D = [(ti, yi), i = 1, 2,..., m) is the line y = x1 +x2t, which minimizesSimilarly, the least squares quadratic (see Example 2) to the data set D is the parabola y = x1 + x2t +x3t2, which minimizes Give a vector space argument to show that E2
The accompanying table is a sample set of seasonal farm employment data (t1. yi) over about a two-year period, where t, represents months and y, represents millions of people. A plot of the data is given in the figure. It is decided to develop a least squares mathematical model of the formy(t) = x1
Find the QR-factorization for each given matrix A.(a)(b)
Find the orthogonal complement of the null space of A.(a)(b)
Let A be an n x n symmetric matrix, and suppose that Rn has the standard Inner product. Prove that if (u. Au) = (u, u) for all u In Rn, then A = In.
An n x n symmetric matrix A is positive semi definite if xT Ax ≥ 0 for all x in Rn. Prove the following: (a) Every positive definite matrix is positive semi definite. (b) If A is singular and positive semi definite, then A is not positive definite. (c) A diagonal matrix A is positive semi
Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R4 with basis
Let A be an n x n skew symmetric matrix. Prove that xT Ax = 0 for all x in Rn.
Let B be an m x n matrix with orthonormal columns b1, b2 ... bn. (a) Prove that m ≥ n. (b) Prove that BT B = In.
Let {u1 ..... uk, uk+1,..., un) be an orthonormal basis for Euclidean space V, S = span {u1......uk}, and T = span (uk+1...un}. For any x in S and any y in T, show that (x, y) = 0.
Let V be a Euclidean space and W a subspace of V. Use Exercise 28 in Section 5.5 to show that (W⊥)⊥ = W.
Show that if A is an m x n matrix such that AAT is non-singular, then rank A = m.
Let u and v be vectors in an inner product space V. If ((u - v), (u + v)) = 0, Show that ||u|| = ||v||.
Let S = {v1, v2,..., vn) be an orthonormal basis for a finite-dimensional inner product space V and let v and w be vectors in V withShow that d(v, w) =
Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R4 with basis.coefficient matrix is not square or is nonsingular. Determine whether this is the case in your software. If it is, compare your software's output with the solution given in Example 1.
Consider the vector space of continuous real-valued functions on [0, π] with an inner product defined by (f, g) = ∫π0 f(t)g (t)dt. Show that the collection of functions sin nt, for n = 1, 2.... is an orthogonal set.
Find the standard matrix representing each given linear transformation. (a) Projection mapping R3 into the xy-plane (b) Dilation mapping R3 into R3 (c) Reflection with respect to the x-axis mapping R2 in R2.
Let A be a fixed 3 x 3 matrix; also let L: M33 → M33 be defined by L(X) = AX - XA, for X in A33. Show that L is a linear transformation.
Let V be an inner product space and let w be a fixed vector in V. Let L: V → R be defined by L(v) = (v, w) for v in V. Show that L is a linear transformation.
Let L: M22 †’ R1 be defined byIs L a linear transformation?
Let V = C [a, b] be the vector space of all real-valued functions that are integrable over the interval [a, b]. Let W = R1 Define L: V → W by L(f) = ∫ab f(x) dx. Prove that L is a linear transformation.
Let A be an nxn matrix and suppose that L: Mnn → Mnn is defined by L(X) = AX, for X in Mnn. Show that L is a linear transformation.
Show that the function L defined in Example 8 is a linear transformation.
Let L: V → W be a linear transformation from a vector space V into a vector space W. The image of a subspace V1 of V is defined as L(V1) = {w in W | w = L(v) for some v in V}. Show that L(V1) is a subspace of V.
Let L1 and L2 be linear transformations from a vector space V into a vector space W. Let {v1, v2 ... vn} be a basis for V. Show that if L1 (v,) = L2 (v) for i = 1, 2 . . . n, then L1 (v) = L2 (v) for any v in V.
Let L: V → W be a linear transformation from a vector space V into a vector space W. The pre-image of a subspace W1 of W is defined as
Let O: Rn → Rn be the zero linear transformation defined by O (v) = 0 for v in Rn. Find the standard matrix representing O.
Let I: Rn → Rn be the identity linear transformation defined by I(v) = v for v in Rn. Find the standard matrix representing.
Complete the proof of Theorem 6.3 by showing that the matrix A is unique. (Suppose that there is another matrix B such that L(x) = Bx for x in Rn. Consider L(ej) for j = 1, 2... n. Show that A = B.)
In Exercise find the standard matrix representing each given linear transformation(a) L: R2†’R2 defined by L(b) L: R2†’R2 defined by L (c) L: R3†’R3 defined by L (u) = ku
Let L: M22 †’ M22 be the linear transformation defined(a) Find a basis for ker L. (b) Find a basis for range L.
Verify Theorem 6.5 for the linear transformation given in Exercise 11.(a) Find a basis for ker L. (b) Find a basis for range L
Let A be an m x n matrix, and consider the linear trans-formation L: Rn → Rm defined by L(x) = Ax, for x in R". Show that Range L = column space of A.
Let L: R5 †’ R4 be the linear transformation defined by(a) Find a basis for and the dimension of ker L. (b) Find a basis for and the dimension of range L.
Let L: V → W be a linear transformation, and let dim V = dim W. Prove that L is invertible if and only if the image of a basis for V under L is a basis for W.
Let L: R3 †’ R3 be defined by(a) Prove that L is invertible. (b) Find
Let L: R2 †’ R2 be the linear operator defined by(a) is In Ker L? (b) is In Ker L? (c) Is In range L? (d) is In range L? (e) Find Ker L (f) Find a set of vectors spanning range L.
Let L: V → W be a linear transformation, and let S = {v1, v2,..., vn} be a set of vectors in V. Prove that if T = {L(v1), L(v2),..., L(vn)} is linearly independent, then so is S. What can we say about the converse?
Let L be the linear transformation defined in Exercise 25, Section 6.1. Prove or disprove the following: (a) L is one-to-one. (b) L is onto.
Let L: Rn Rm be a linear transformation defined by L(x) = Ax, for x in R". Prove that L is onto if and only if rank A = m.
Let L: P1→ P2 be defined by L(p(t)) = tp(t) + p(0). Consider the ordered bases S = [t, 1} and S' - {t + 1, t - 1} for P1, and T = [t2, t, 1} and T' = {t2 + 1, t - 1, t + 1} for P2. Find the representation of L with respect to (a) S and T (b) S' and T'. (c) Find L(-3t - 3) by using the definition
Let L: V †’ V be a linear operator. A nonempty sub-space U of V is called invariant under L if L(U) is contained in U. Let L be a linear operator with invariant subspace U. Show that if dim U = m and dim V = n, then L has a representation with respect to a basis S for V of the formWhere A
If L: R3 †’ R2 is the linear transformation whose representation with respect to the natural bases for R3 and R2, isFind each of the following: (a) L ([1 2 3]) (b) L ([-1 2 -1]) (c) L ([0 1 2]) (d) L ([0 1 0]) (e) L ([0 0 1])
If O: V → W is the zero linear transformation, show that the matrix representation of O with respect to any ordered bases for V and W is the m x n zero matrix, where n = dim V and m = dim W.
If I : V → V is the identity linear operator on V defined by I (v) = v for v in V, prove that the matrix representation of I with respect to any ordered basis S for V is In where dim V = n.
Let L: R4 †’ R3 be defined byL([u1 u2 u3 u4]) = [u1 u2+u3 u3+ u4]Let S and T be the natural bases for R4 and R3, respectively. LetAnd (a) Find the representation of L with respect to S and T. (b) Find the representation of L with respect to S' and T' (c) Find L ([2 1 -1 3]) by using the
Let V be the vector space of real-valued continuous functions with ordered basis S = {sint, cost} and consider T = {sint - cost, sint + cost}, another ordered basis for V. Find the representation of the linear operator L: V → V defined by L(f) = f' with respect to (a) S; (b) T; (c) S and T; (d) T
Let L: V → V be a linear operator defined by L(v) = cv, where c is a fixed constant. Prove that the representation of L with respect to any ordered basis for V is a scalar matrix.
Let the representation of L: R3 †’ R2 with respect to the ordered bases S = [v1, v2, v3} and T = [w1, w2) beWhere (a) Compute [L(v1)]T [L(v2)]T and [L(v3)]T. (b) Compute L (v1), L (v2), and L (v3). (c) Compute
Let I: V → V be the identity operator on an n-dimensional vector space V and let S = {v1, v2,..., vn) and T = {w1, w2,..., wn} be ordered bases for V. Show that the matrix of the identity operator with respect to S and 7 is the transition matrix from the 5-basis to the T-basis.
Let L: M22 †’ M22 be defined byFor A in M22. Consider the ordered bases And For M22 find the representation of L with respect to (a) 5; (b) T; (c) S and T (d) T and S.
Let L1, L2, and L3 be linear transformations of R3 into R2 defined byProve that S = {L1, L2, L3} is a linearly independent set in the vector space U of all linear transformations of R3 into R2.
Find two linear transformations L1: R2 → R2 and L2: R2 → R2 such that L2 o L1 ≠ L1 o L2.
Find a linear transformation L: R2 → R2, L ≠ O, the zero transformation, such that L2 = O.
Let U be the set of all linear transformations of V into W, and let O: V → W be the zero linear transformation defined by O(x) = 0W for all x in V.(a) Prove that O ⊞ L = L E ⊞ 0 = L for any L in U.(b) Show that if L is in U, thenL⊞ ((-l) ⊡ L) = O.
Let L: V → V be a linear transformation represented by a matrix A with respect to an ordered basis S for V. Show that A2 represents L2 = L o L with respect to 5. Moreover, show that if k is a positive integer, then Ak represents Lk = L o L o . . . o L(k times) with respect to S.
Verify that the set U of all linear transformations of V into W is a vector space under the operations ⊞ and ⊡.
Let V1, V2, and V3 be vector spaces of dimensions n, m, and p, respectively. Also let L1: V1 → V2 and L2: V2 → V3 be linear transformations. Prove that L2 o L1: V1 → V3 is a linear transformation.
Let L1, L2, and S be as in Exercise 3. Find the following: (a) (L1 o L2) ([u1 u2 u3]) (b) (L2 o L1) ([u1 u2 u3]) (c) The representation of L1 o L2 with respect to S (d) The representation of L2 o L1 with respect to S
Let A, B, and C be square matrices. Show that (a) A is similar to A. (b) If B is similar to A, then A is similar to B. (c) If C is similar to B and B is similar to A, then C is similar to A.
Let L be the linear transformation defined in Exercise 2, Section 6.3. (a) Find the transition matrix P from S' to S. (b) Find the transition matrix from S to S' and verify that it is P-l. (c) Find the transition matrix Q from T' to T. (d) Find the representation of L with respect to S' and T'. (e)
Show that if A and B are similar matrices, then Ak and Bk are similar for any positive integer k.
Show that if A and B are similar, then AT and BT are similar.
Prove that if A and B are similar, then Tr (A) = Tr(B)
Determine the matrix in homogeneous form that produced the image of the semicircle depicted in the following figure:
Determine the matrix in homogeneous form that produced the image of the semicircle depicted in the following figure:
Let the columns ofbe the homogeneous form of the coordinates of the vertices of a triangle in the plane. Note that the first and last columns are the same, indicating that the figure is a closed region. (a) In a coordinate plane, sketch the triangle determined by S. Connect its vertices with
A plane figure S is to be translated byAnd then the resulting figure translated by (a) Determine the 3 x 3 matrix M in homogeneous form that will perform this composition of translations, (b) Can the transformations be reversed? That is, is it possible to determine a matrix P that will return the
Let A be the 3 x 3 matrix in homogeneous form that translates a plane figure byAnd let B be the 3 x 3 matrix in homogeneous form that translates a plane figure by the vector Will the image be the same regardless of the order in which the transformation are performed? Explain your answer
Determine the matrix in homogeneous form that produced the image of the triangle depicted in the following figure:
For an n x n matrix A, the trace of A, Tr(A), is defined as the sum of the diagonal entries of A. Prove that the trace defines a linear transformation from Mnn to the vector space of all real numbers.
Let L1: V → V and L2: V → V be linear transformations on a vector space V. Prove that (L1 +L2)2 = L2 + 2L1 o L2 + L22 if and only if L1 o L2 = L2 o L1.
Let u and v be nonzero vectors in R". In Section 5.3 we defined the angle between u and v to be the angle Î¸ such thatA linear operator L: Rn †’ Rn is called angle preserving if the angle between u and v is the same as that between L(u) and L(v). Prove that if L is inner
Let L: V → W be a linear transformation. If {v1, v2,..., vk} spans V, show that {L(v,), L(v2),..., L(vk)} spans range L.
Let L: M22 → M22 be the linear transformation defined by L(A) = AT. (a) Find a basis for ker L. (b) Find a basis for range L.
Let A be a 3 ( 3 matrix with first row [k 0 0] for some nonzero real number k. Prove that k is an eigenvalue of A?
Let L: P2 → P2 be defined by L(at2 + bt + c) = ct + b. (a) Find the matrix representing L with respect to the basis S = (t2, 2 + t, 2 - t}. (b) Find the eigenvalues and associated eigenvectors of L?

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