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linear algebra

Linear Algebra With Applications 7th Edition W. Keith Nicholson - Solutions

If A and B are positive definite and r > 0, show that A + B and rA are both positive definite. If A and B are positive definite, show that
If A is an n × n positive definite matrix and U is an n × m matrix of rank m, show that UTAU is positive definite.
In each case find the QR-factorization of A.(a)(b)
If R is upper triangular and invertible, show that there exists a diagonal matrix D with diagonal entries ±1 such that Rl = DR is invertible, upper triangular, and has positive diagonal entries.
In each case, find the exact eigenvalues and determine corresponding eigenvectors.(a) (b)
In each case, find the exact eigenvalues and then approximate them using the QR-algorithm.(a)(b)
If A is symmetric, show that each matrix Ak in the QR-algorithm is also symmetric. Deduce that they converge to a diagonal matrix.
In each case, compute the norm of the complex vector. (a) (1 - i, l + i, 1, -1) (b) (-2, -i, 1, + i, 1 - i, 2i)
Prove Theorem 2.Let (e1, e2, ..., en} be an orthogonal set of vectors.1. {r1, e1, r2e2,..., rnen} is also orthogonal for any r, ‰ 0 in R.2.
Show that the diagonal entries of any hermitian matrix are real.
A complex matrix S is called skew-hermitian if SH = -S. (a) If S is skew-hermitian, show that S2 and iS are hermitian. (b) Show that every n x n complex matrix Z can be written uniquely as Z = V + S, where V is hermitian and S is skew-hermitian.
Show that the inverse of a unitary matrix is again unitary.
Show that (a) fails if normal is replaced by hermitian.
In each case, determine whether the two vectors are orthogonal. (a) (i, - i, 2 + i), (i, i, 2 - i) (b) (4 + 4i, 2 + i, 2i), (-1 + i, 2, 3 - 2i)
If below amount show that U-lAU is not upper triangul for any real invertible matrix U.
A subset U of Cn is called a complex subspace of Cn if it contains 0 and if, given Z and W in U, both Z + W and zZ lie in U (z any complex number). In each case, determine whether U is a complex subspace of
In each case, find a basis over C, and determine the dimension of the complex subspace U of C3 (see the previous exercise). (a) U = {(iv + w, 0, 2v - w) | v, w in C} (b) U = {(u, v, w) | 2u + (1 + i)v - iw = 0; u, v, w in C}
In each case, determine whether the given matrix is hermitian, unitary, or normal.(a)(b) (c) (d) (e) (f) (g) (h)
In each case, find a unitary matrix U such that UHZU is diagonal.(a)(b) (c) (d) (e) (f)
Find all a in Z10 such that: (a) a has an inverse (and find the inverse) (b) a = 2k for some k ≥ 1
(a) If a binary7 linear (n, 3)-code corrects two errors, show that n ‰¥ 9.(b) ifShow that the binary (10, 3) - code generated by G corrects two errors. [ It can be shown that no binary (9, 3)-code corrects two errors.]
Show that 2a - 0 in Z10 holds in Z10 if and only if a = 0 or a = 5.
Find the inverse of: 11 in Z19.
In each case show that the matrix A is invertible over the given field, and find A-1.(a)(b)
Consider the linear system 3x + y + 4z = 3 4x + 3y + z = l In each case solve the system by reducing the augmented matrix to reduced row-echelon form over the given field: Z7.
Let K be a vector space over Z3 with basis {1, t}, so K = {a + bt | a, b in Z3}. It is known that K becomes a field of nine elements if we define t2 = -1 in Z3. In each case find the inverse of the element x of K: x = 1 + t.
In each case, find a symmetric matrix A such that q = XTBX takes the form q = XTAX.(a)(b) (c) (d)
In each case, find a change of variables that will diagonalize the quadratic form q. Determine the index and rank of q. (a) q = x21 + 2x1x2 + x22 (b) q = x21 + 4x1x2 + x22 (c) q = x21 +x22 +x23 - 4(x1x2 + x1x3 + x2x3) (d) q = 7x21 + x22 + x23 + 8x1x2 + 8x1x3 -16x2x3 (e) q = 2(x21 + x22 + x23 -
For each of the following, write the equation in terms of new variables so that it is in standard position, and identify the curve. (a) xy = 1 (b) 3x2 - 4xy = 2 (c) 6x2 + 6xy - 2y2 = 5 (d) 2x2 + 4xy + 5y2 = 1
Consider the equation ax2 + bxy + cy2 = d, where b ‰ 0. Introduce new variables x1 and y1 by rotating the axes counterclockwise through an angle Î¸. Show that the resulting equation has no xy1-term if Î¸ is given by
If X = [x1 ∙ ∙ ∙ xn]T is a column of variables, A = AT is n × n, B is 1 × n, and c is a constant, XTAX + BX = c is called a quadratic equation in the variables xj. (a) Show that new variables y1,... ,yn can be found such that the equation takes the form λ1y21 + ∙ ∙ ∙ + λry2r + k1y1
In each case, find the coordinates of v with respect to the basis B of the vector space V.(a) V= P2, v = 2x2 + x - 1, B = {x + 1, x2, 3}(b) V- P2, v = ax2 + bx + c,B = {x2, x+ 1, x + 2}(c) r = R3,v = (l,- l,2),B = {(1,-1,0),(1, 1, 1), (0, 1, 1)}(d) V= R3, v = (a,b,c),B = {(1,-1, 2), (1,1,-1),
Let T : V → W be an isomorphism, let B = [e1.........,en] be an ordered bases of V, and let D = [T(e1),.........T(en)]. Show that MDB (T) = 1n - the n × n identity matrix.
Let T: P2 †’ R3 be defined byT(p) = (p(0)), p(1),(p(2)) for all p in P2. LetB = {1,x, x2}and D = [(1,0,0),(0,1,0),(0,0,1)](a) show that MDB (T) =And conclude that T is an isomorphism. (b) Generalize to T: Pn †’ Rn+1 where T(P) = (P(a0) p(a1..., p(an)) and a0, au...,a" are
Suppose T: P2 †’ R2 is a linear transformation. If B = {1, x, x2} and D = {(1, 1), (0, 1)}, find the action of T given:(a )MDB(T) =(b) MDB(T) =
Show that the following properties hold provided that the transformations link together in such a way that all the operations are defined.(a) R(ST) = (.RS)T(b) 1wT = T = T1y(c) R(S + T) = RS + RT(d) (S + T)R = SR + TR (e) (aS)T = a(ST) = S(aT)
Given S and T in L(V, W), show that: (a) ker S ∩ ker T ⊂ ker(S + T) (b) im(S + T) ⊂ imS + im T
Let V and W be vector spaces. If X is a , subset of V, defineX° = [T in L(V, W) | T(v) = 0 for all v in X)(a) Show that X° is a subspace of L(F, IV).♦ (b) If Xcxu show diat X°.(c) If U and U\ are subspaces of V, show that (U+ Ul)°=U°n l/f.
In each case, find the matrix of T: V †’ W If corresponding to the bases B and D of V and IF, respectively.(a)T: M22 †’ R, T(A) = tr A;D = {1} (b) T: M22 †’ M22 , T(A) = AT; (c) T: P2 †’ P3, T|p(x) = xp(x); B = [1,x,x2] D = [1,x,x2,x3] (d) T: P2
In each case, find the matrix of T:V †’ W corresponding to the bases B and D, respectively, and use it to compute CD[T(v)], and hence T(v).a. T: R3 †’ R4,T(x,y,z) = (x + z,2z,y - z, x + 2); B and D standard; v = 1,/-1,3b. T: R2 †’ R4, T(x,y) = (2x - y, 3x + 2y, 4y,x);B
In each case, verify Therorem 3. Use the standard basis in Rn and [1,x,x2] in P2a.b. c. d.
In each case, find T-1 and verify that MDB (T)-1 = MBD(T-1).(a) T: R2 → R2, T(a,b) = (a + 2b, 2a + 5b); B = D = standard(b) T: R3 → R3 T(a,b,c) = b + c, a + c, a + b); B = D = standard(c) T : P2 → R3,T(a + bx + cx2) = a - c,b,2a - cB = [1,x, x2], D = standard(d) T : P2 → R3,T(a + bx + cx2)
In each case, show that MDB (T) is invertible and use the fact that MDB (T-1) = [MBD(T)]-1 to determine the action of T-1(a) T: P2 †’ R3, T(a + bx + cx2) = a + c, c, b - c);B = {1,x, x2}, D = standard(b) T : M22 †’ R4.B =D = standard
In each case find PD † b, where B and ordered bases of V. Then verily that CD(v) = PD † BCBW-(a) F= R2, B = f(0,-l),(2, 1)}D = ((0, 1),(1, I)}, v = (3,-5)™¦ (b) V= P2, B = [1 + x + x2},D = {2, x + 3, x2 - 1}, v = 1 + x + x2(c) V = M22,
If A and B are n x n matrices, show that they have the same null space if and only if A = UB for some invertible matrix U. [Hint: Exercise 28 §7.3.]
Given a complex number w, define TU: C → C by Tz ,(z) = wz for all z in C.(a) Show that Tu, is a linear operator for each w in C, viewing C as a real vector space.(b) If B is any ordered basis of C, define S: C -> M22 by S(w) = Mb(Tu.) for all w in C. Show that S is a one-to-one linear
In each case verily that PD ← b is the inverse of PB ← D and that PE ← DPD ←B = PE ← B, where B, D, and E are ordered bases of V. (a) K = R3, fl = {(l, 1, 1), (1, -2, 1), (1, 0, -1)), D = standard basis, E = {(1, 1, 1), (1,-1, 0), (-1, 0, 1)} (b) V = P2, B = {1, x, x2}, D = {1 + x + x2, 1
Use property (2) of Theorem 2, with D the standard basis of IR", to find the inverse of:(a)(b)
In each case, find P = P = B† B and verify that P-1MB0(T)P = MB(T) for the given operator T.(a) T :R3 †’R3, T(a, b, r) = (2a -b ,b + c, c - 3a); B0 = {(1, 1,0), (1,0, 1), (0, 1,0)} and B is the standard basis.(b) T: P2 † P2,T(a + bx + cx2) = (a + b) + (b + c)x + (r + a)x2; B0 = {1, x,
In each case, verify that P-1AP = D and find a basis B of R2 such that MB(TA) = D.(a)(b)
In each case, compute the characteristic polynomial cT(x).(a) T : R2 †’ R2, T(a, b) = (a - b, 2b - a)(b) T: R2 †’ R2, T(a, b) = (3a + 5b, 2a + 3b)(c) T: P2 †’ P2, T(a + bx + cx2) = (a - 2c)+ (2a + b + c)x + (c - a).x2(d) T: P2 †’ P2, T(a + bx + cx2) = {a + b - 2e)+ (a- 2b + r)x + {b
In each case, show that V = U Š• W.(a) V= R4, U= span{(l, 1, 0, 0), (0, 1, 1, 0)},W= span{(0, 1,0, 1), (0, 0, 1, 1)}(b) V- R4, U = {(tf, b, b) | a, b in R},W = {{c,d,c,-d) | c,dm R}(c) V = P3, U = [a + bx | a ,b in R}W = [ax2 + bx3 | a, b in R](d) V = M22,U
Let E be a 2 × 2 matrix such that E2 = E. Show that M22 = U ⊕ W, where U = [A | AE = A} and W = [B | BE = 0}. [Hint: XE lies in U for every matrix X.]
Let U and W be subspaces of V, let dim V = n and assume that dim U + dim W = n. (a) If U ∩ W= {0}. show that V = U⊕W. (b) If U+W= V,, show that V = U⊕W. [Hint: Theorem 5 §6.4.]
Let Aand consider TA : R2 †’ R2.(a) Show that the only eigenvalue of TA is A = 0.(b) Show that ker(TA) = (R)is the unique TA-invariant subspace of R2: (except for 0 and R2).
If T: V → V is any linear operator, show that ker T and im T are T-invariant subspaces.
Let T: V → V be a linear operator where dim V = n. If U is a T-invariant subspace of V, let T1 : U → U denote the restriction of T to U (so Tj(u) = T(u) for all u in U). Show that c T(x) = cTI(x) ∙ q(x) for some polynomial q(x). [Hint: Theorem 1.]
In each case, show that T-2 = 1 and find (as in Example 10) an ordered basis B such that Mb(T) has the given block form.(a) T: M22 †’ M22 where T(A) = AT,(b) T : P3 †’ P1 where 7[p(x)] = p(-x), (c) T : C †’ C where T{a + bi) = a - bi, (d) T: IR3 †’ IR3 where
Let U and W denote subspaces of a vector space V.(a) If V = U ⊕ W, define T : V→ V by T(v) = w where v is written (uniquely) as v = u + w with u in U and w in W. Show that 7 is a linear transformation, U = ker T,W= im T, and T2 = T.(b) Conversely, if T : V → V is a linear transformation such
In each case, show that T2 = Tand find (as in the preceding exercise) an ordered basis B such that MB(T) has the form given (0)k is the k Ã— k zero matrix).(a) T: P2 †’ Pi where T(a + bx + cx2) ={a - b + c)(l + x + x2), MB(T) =(b) T:R3 †’ R3where T{a, b, c) =(a + 2b, 0,4b + c), MB(T)
Let V be a vector space. If f: V → [R is a linear transformation and z is a vector in V, define Tj z: V → V by Tf.z(v) = f(v)z for all v in V. Assume that f ≠ 0 and z ≠ 0. (a) Show that Tj z is a linear operator of rank 1. (b) If ≠ 0, show that 7f . z is an idempotent if and only if f(z)
Let S and T be linear operators on V and assume that ST = TS.(a) Show that imS and kerS1 are T-invariant.(b) If U is T-invariant, show that S(U) is T-invariant.
Let U be a fixed n × n matrix, and consider the operator T: Mnu → M"" given by T(A) = UA.(a) Show that A is an eigenvalue of T if and only if it is an eigenvalue of U.(b) If A is an eigenvalue of T, show that EX(T) consists of all matrices w hose columns lie in Eλ(U): Eλ(T) = ]P1, P2 - Pn][P,
Show that the only subspaces of V that are T-invariant for every operator T : V → V are 0 and V. Assume that V is finite dimensional. [Hint: Theorem 3 §7.1.1
In each case, show that U is T-invariant, use it to find a block upper triangular matrix for T, and use that to compute c1(x).(a) T: P2 → P2, T{a + bx + cx2) = (-a + 2b + c) + (a + 3b + c)x + {a + 4b)x2, U = span{l, x + .v}(b) T : P2 → P2, T{a + bx + cx2) = (5a - 2b + c)+ (5a - b + c)x + {a +
In each case, show that TA: B2 †’ IR' has no invariant subspaces except 0 and R2.(a)(b)
In each case, determine which of axioms P1-P5 fail to hold. (a) V = R2, ((x1,y1),(x2, y2)) = x1y1x2y2 (b) V = R3 , ((x1, x2, x3),(y1,y2,y3)) = x1y1 -x2y2 + x3y3 (c) V = IR, (z,w) = zw, where w is complex conjugation (d) V = P3(p(x), q(x)) = p(l)q(l) (e) V = M22, (A, B) = det (AB) (f) V = F[0,1],
In each case, show that (v, w) = vTAw defines an inner product on R2 and hence show that A is positive definite(a)(b)(c)(d)
In each case, find a symmetric matrix A such that (v, w) = vTAw.(a)(b)(c)(d)
If A is symmetric and XTAX = 0 for all columns X in R", show that A = 0. [Hint: Consider (X +Y, X + Y) where (X, Y) = XTAY. ]
Let ||u|| = 1, ||v|| = 2, || w|| = V3,
Verify that the dot product on R" satisfies axioms P1-P5.
Let u and v be vectors in an inner product space V. (a) Expand (2u - 7v, 3u + 5v). (b) Expand (3u - 4v, 5u + v). (c) Show that ||u + v||2 = ||u||2 + 2(u,v) + ||v||2. (d) Show drat ||u - v||2 = ||u||2 - 2(u,v) + ||v||2.
Let v denote a vector in an inner product space V. (a) Show that W = (w | w in V,(v, w) = 0} is a subspace of V. (b) If V= R3 with the dot product, and if v = (1, -1, 2), find a basis for IV.
If V = span{v1,v2,...,v"} and (v,v,) = (w,v,) holds for each i. Show that v = w.
Use the Cauchy-Schwarz inequality in an inner product space to show that:(a) If | u || < 1, then (u,v)2 < ||v||2 for all v in V.(b) (x cosθ + y sinθ) y2 < x2 + y2 for all real x,y, and θ.(c) ||r1v1 + - + rnvn||2 < [r1 ||+ - + rn|| v||]2 for all vectors v" and all r, > 0 in R.
In each case, find a scalar multiple of v that is a unit vector.(a) v = f in C[O, 1] wheref(x) = x2(b) v = fin C[-Ï€. Ï€ ] where f(x) = cos x(c)(d)
In each case, find the distance between u and v. (a) u = (3, -1, 2, 0), v = (1, 1, 1, 3) (b) u = (l, 2, -1, 2), v = (2, 1, -1, 3) (c) u = f1v = g in C[0, 1] where f(x) = x2 and g(x) = 1 - x (d) u = f v = g in C[-π, π] where f(x) = 1 and g(x) = cos x
Let Dn denote the space of all functions from the set {1, 2, 3,..., n} to IR with point-wise addition and scalar multiplication (see Exercise 35 §6.3). Show that (,) is an inner product on Dn if (f ∙ g) = f(1)g(D + f(2)g(2) +∙∙∙∙∙∙∙∙∙+ f(n)g(n)
Use the dot product in R" unless otherwise instructed.In each case, verify that B is an orthogonal basis of V with the given inner product and use the expansion theorem to express v as a linear combination of the basis vectors(a)(b) V = R3, (v, w) = VTAw where A = (c) v = a + bx + cx2, B = {1, x, 2
Let v and w be vectors in an inner product space V. Show that:(a) v is orthogonal to w if and only if ||v + w|| = ||v-w||.(b) v + w and v - w are orthogonal if and only if ||v|| - ||w||.
If X is any set of vectors in an inner product space V, defineX⊥ = {v | v in V, (v, x) = 0 for all x in X}.(a) Show that X⊥ is a subspace of V.(b) If U = span, (u1,u2..., um}, show that (U⊥ = {u1,...,um.(c) If X⊂ Y, show that Y⊥ ⊂ X⊥.(d) Show that X⊥ ∩ Y⊥ = (X ∪ Y) ⊥
Let U be a finite dimensional subspace of an inner product space V, and let v*be a vector in V.(a) Show that v lies in U if and only if v = projtU(v).(b) If V= R3, show that (-5, 4, -3) lies in span{(3, -2, 5), (-1, 1, 1)} but that (-1, 0, 2) does not.
Let; n ‰ 0 and w ‰ 0 he nonparallel vectors in IR? (as in Chapter 4).(a)orthogonal basis of R3.(b)plane through the origin with normal n-.
Let R3 have the inner product((x, y, z), (x΄, y΄, z΄)) = 2xx΄ + yy' + 3zz. In each case, use the Gram-Schmidt algorithm to transform B into an orthogonal basis.(a) B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}(b) B = {(1, 1, 1), (1, -1, 1), (1, 1, 0)}
Let v be a vector in an inner product space V.(a) Show that ||v|| > ||projU(v)|| holds for all finite dimensional subspaces U.[Hint: Pythagoras' theorem.]If {e1,e2,..., ew} is any orthogonal set in V, prove Bessel's inequality:
Let M22 have the inner product(X, Y) = tr(xyT). In each case, use the Gram-Schmidt algorithm to transform B into an orthogonal basis.(a)(b)
In each case, use the Gram-Schmidt process to convert the basis B = (1, x, a2} into an orthogonal basis of P2. (a) (p, q) = p(0)q(0) + p(1)q(l) + p(2)q(2) (b) (p, q) = ∫20p(x)q(x) dx
In each case find U-1 and compute dim U1 and dim U1(a) U = span{[l 1 2 0], [3 -1 2 1],[1 -3 -2 1]} in R4(b) U = span{[l 1 0 0]} in R4(c) U = span{l, x} in P2 with(P. q) = P(0)q(0) + p(1)q(1) + p(2)q(2)(d) U = span{x} in P2 with(p, q) = ˆ«l0p(x)q(x)dx(e)(f)
Let (X, Y) = ti'(XYT) in M22. In each case find the matrix in U closest to A(a)(b)
Let (p(x), q(x)) = p(0)q(0) + p(1)q(1) + p(2)q(2) in P2. In each case find the polynomial in U closest to f(x). (a) U = span{l + x, x2}, f(x) = 1 + x2 (b) U = span{l, 1 + x2}; f(x) = x
Using the inner product (p, q) = ∫10p(x)q(x) dx on P2, write v as the sum of a vector in U and a vector in UL.(a) v = x2, LJ = span{x + 1, 9x - 5}(b) v = x2 + 1, U = span{ 1, 2x - 1}
In each case, show that T is symmetric by calculating MB(T) for some orthonormal basis B.(a) T:R3 †’ R3;T(a, b, c) = (a-2b, -2a + 2b + 2c, 2b-c); dot product(b) T:M22 †’ M22;(c) T: P2 †’ P2; T(a + bx + cx2) = (b + c) + (a + c)x + (a + b)x2; inner product (a + bx +
Let T: V → Vbe a linear operator on an inner product space V of finite dimension. Show that the following are equivalent. (1) (v, T(w)) = -(T(v), w) for all v and w in V. (2) MB(T) is skew-symmetric for every orthonormal basis B. (3) MB(T) is skew-symmetric for some orthonormal basis B. Such
Let B = {e1, e2,..., en} be an orthonormal basis of an inner product space V. Given T: V †’ V, define T': V †’ V by(a) Show that (aT)' = aT'.(b) Show that (5 + T)' = S' + T'(c) Show that MB(T') is the transpose of MB(T).(d) Show that (T')' = T, using part (c).[Hint: Mb(S) = MB(T) implies
Let Vbc an w-dimensional inner product space, and let T and S denote symmetric linear operators on V. Show that:(a) The identity operator is symmetric.(b) rT is symmetric for all r in R.(c) S + T is symmetric.(d) If T invertible, then T-1 is symmetric.(e) If ST = TS, then ST is symmetric.
In each case, show that T is symmetric and find an orthonormal basis of eigenvectors of T.(a) T:(R3→R3; T(a, b, c) = (2a + 2r, 3b, 2a + 5c)-, use the dot product(b) T: R3 →R3; T(a, b, c) = (7a - b, -a + 7,b,2c); use the dot product(c) T: P2 → P2 T(a + bx + cx2) = 3b + (3a + 4c)x'+4bx2 inner
Let T: M22 -> M22 be given by T(X) = AX, where A is a fixed 2 Ã— 2 matrix.(a) Compute MB(T), whereNote the order! (b) Show that cT(x) = [CA(x)]2. (c) If the inner product on M22 is (X, Y) = tr(XYT), show that T is symmetric if and only if A is a symmetric matrix.
Let T: R2†’ R2be given by T(a, b) = (b €“ a, a + 2b). Show that T is symmetric if the dot product is used in R2but that it is not symmetric if the following inner product is used:
In each case, show that T is an isometry of IR", determine whether it is a rotation or a reflection, and find the angle or the fixed line. Use the dot product.(a)(b)(c)(d)(e)(f)

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