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Questions and Answers of Linear Algebra

If U and Ware subspaces of V, define their intersection U ∩ W as follows: U ∩ W = {v|v is in both U and W} (a) Show that U ∩ W is a subspace contained in L and W. (b) Show that U ∩ W {0} if
A polynomial p(x) is even if p(- x) = p(x) and odd if p(- x) = - p(x). Let En and On denote the sets of even and odd polynomials in Pn. Show that On is a subspace of Pn and find dim On.
Find all values of x such that the following are independent in R3. (a) {(2, x, 1), (1, 0, 1), (0, 1, 3)}
Show that the following are bases of the space V indicated. (a) {(- 1, 1, 1), (1, - 1, 1), (1, 1, - 1)}; V = R3 (b) {1 + x, x + x2, x2 + x3, x3}; V = P3
Exhibit a basis and calculate the dimension of each of the following subspaces of P2. (a) {a + b(x + x2) | a and b in R} (b) {p(x) | p(x) = p(- x)}
Exhibit a basis and calculate the dimension of each of the following subspaces of M22.(a)(b)
Find a basis of U not containing ALetand define U = {X | X is in M22 and AX = X).
In each case, find a basis for V that includes the vector v. (a) V = R3, v = (0, 1, 1) (b) V = P2, v = x2 - x + 1
If A is a square matrix, show that det A = 0 if and only if some row is a linear combination of the others.
Let D, I, and X denote finite, nonempty sets of vectors in a vector space V. Assume that D is dependent and / is independent. In each case answer yes or no, and defend your answer. (a) If X ⊆ D,
Given v1, v2, v3,..., vk, and v, let U = span{v1, v2,..., vk} and W = span{v1,..., vk, v}. Show that either dim W = dim U or dim W = 1 + dim U.
Let U and W be subspaces of a vector space V. (a) If dim V = 3, dim U = dim W = 2, and U ≠ W, show that dim(U ∩ W) = 1. (b) Interpret (a) geometrically if V= R3.
In each case, find a basis for V among the given vectors. V= P2, {x2 + 3, x + 2, x2 - 2x - 1, x2 + x)
Let V be the set of all infinite sequences(a0, a1, a2,...) of real numbers. Define addition and scalar multiplication by (a0, a1,...) + (b0, b1,...) = (a0 + b0, a1 + b1,...) and r(a0, a1,...) = (ra0,
Let U and W be subspaces of V (a) Show that U + W is a subspace of V containing U and W. (b) Show that span {u, w} = Ru + Rw for any vectors u and w. (c) Show that span {u1,...um, w1...,wn} = span{u1
In each case, find a basis of v containing v and w. (a) V = (R4, v = (0, 0, 1, l),w = (1, 1, 1, 1) (b) V= P3, v = x2 + 1, w = x2 + x
If z is neither real nor pure imaginary, show that {z, ) is a basis of C.
Find a basis of P3 consisting of polynomials whose coefficients sum to 4. What if they sum to 0?
If {u, v, w} is a basis of V, determine which of the following are bases. (a) {2u + v + 3w, 3u + v - w, u - 4w) (b) {u, u + w, u - w, v + w}
Can four vectors span R3? Can they be linearly independent? Explain.
Let a and b denote distinct numbers. Show that {(x - a)2, (x - a)(x - b), (x - b)2} is a basis of P2.
Let a and b be two distinct numbers. Assume that n ≥ 2 and let Un = {f(x) in Pn | f(a) = 0 = f(b)}. Show that dim Un = n - 1.
Expand each of the following as a polynomial in powers of x - 1. (a) f(x) = x3 + x + 1 (b) f(x) = x3 - 3x2 + 3x
Use Theorem 2 to show that the following are bases of P2. (a) {x2 - 3a + 2, x2 - 4x + 3, x2 - 5x + 6}
Find the Lagrange interpolation expansion of f(x) relative to a0 = 1, a1 = 2 and a2 = 3 if: (a) f(x) = x2 + x + 1
Find a solution f to each of the following differential equations satisfying the given boundary conditions. (a) fʹ + f = 0; f(1) = 1 (b) fʹʹ + fʹ- 6f = 0; f(0) = 0, f(1) = 1 (c) fʹʹ - 4fʹ + 4f
Find the general solution to fʹ + f = 2.
Consider the differential equation fʹʹ + afʹ + bf = g, where g is some fixed function. Assume that f0 is one solution of this equation, Find a solution to fʹʹ + fʹ - 6f = 2x3 - x2 - 2x.
The population N(t) of a region at time t increases at a rate proportional to the population. If the population doubles in 5 years and is 3 million initially, find N(t).
As a pendulum swings (see the diagram), let t measure the time since it was vertical. The angle Î¸ = Î¸(t) from the vertical can be shown to satisfy the equation
Let [v1, v2,...,vn} be a basis of Rn (written as columns), and let A be an n × n matrix. If {Av1, Av2,..., Ayn} is a basis of Rn, show that A is invertible.
Show that null A = null(AT A) for any real matrix A.
Show that each of the following functions is a linear transformation. (a) T: R2 → R2; T(x, y) = (x, -y) (reflection in the X axis) (b) T: R3 → R3; T(x, y, z) = (x, y, -z) (reflection in the X-Y
Describe all linear transformations T: R → V.
Let T: V → W be a linear transformation. (a) If U is a subspace of V, show that T(U) = (T(u) | u in U} is a subspace of W (called the image of U under T). (b) If P is a subspace of W, show that {v
Suppose T: V → V is a linear operator with the property that T[T(v)] = v for all v in V. (For example, transposition in Mnn or conjugation in C.) If v ≠ 0 in V, show that (v, T(v)) is linearly
In each case, show that T is not a linear transformation. (a) T: Mn,n → R; T(A) = det A (b) T: Mn,n → R; T(A) = rank A (c) T: (R → R; T(x) = x2 (d) T: V → V; T(v) = v + u where u ≠ 0 is a
Given a in R, consider the evaluation map Ea: Pn → R defined in Example 3. (a) Show that Ea is a linear transformation satisfying the additional condition that Ea(xk) = [Ea(xk)] = holds for all k =
In each case, assume that T is a linear transformation.(a) If T: V †’ R and T(v1) = 1, T(v2) = -1, find T(3v1 - 5v2).(b) If T: †’ (R and T(v1) = 2, T(v2) = -3, find T(3v1 +
In each case, find a linear transformation with the given properties and compute T(v).(a) T: R2 †’ R3; T(1, 2) = (1, 0, 1), T(-l, 0) = (0, l, l); v = (2, 1)(b) T: R2 †’ R3; T(2,
If T: V → V is a linear transformation, find T(v) and T(w) if: (a) T(v + w) = v - 2w and T(2v - w) = 2v (b) T(v + 2w) = 3v - w and T(v - w) = 2v - 4w
Let {v1,..., vn} be a basis of V and let T: V → V be a linear transformation. (a) If T(v1) = v1 for each i, show that T = 1v. (b) If T(v1) = -v1 for each i, show that T = -1 is the scalar operator
For each matrix A, find a basis for the kernel and image of TA, and find the rank and nullity of TA.(a)(b) (c) (d)
Let T: Mnn → R denote the trace map: T(A) = tr A for all A in Mnn. Show that dim(ker T) n2 - 1
Let A and B be m × n and k × n matrices, respectively. Assume that AX = 0 implies BX = 0 for every n-column X. Show that rank A > rank B.
Define T: Pn → R by T[p(x)] = the sum of all the coefficients of p(x). (a) Use the dimension theorem to show that dim(ker T) = n. (b) Conclude that (x - 1, x2 - 1,..., xn - 1} is a basis of ker T.
In each case, (i) find a basis or ker T, and (ii) find a basis of m T.(a) T: P2 †’ R2; T(a + bx + cx2) = (a, b)(b) T: P2 †’ R2; T(p(x)) = (p(0), p(1))(c) T: R3 †’ R3;
Let U and V denote the spaces of symmetric and skew-symmetric n × n matrices. Show that dim U + dim V = n2
Fix a column Y ≠ 0 in Rn and let U = [A in Mnn | AY = 0}. Show that dim U = n(n - 1).
Let U be a subspace of a finite dimensional vector space V. (a) Show that U = ker T for some linear transformation T: V → V. (b) Show that U = im S for some linear transformation S: V → V.
Let P: V → R and Q : V → R be linear trans-formations, where V is a vector space. Define T: V → R2 by T(v) = (P(v), (Q(v)). (a) Show that T is a linear transformation. (b) Show that ker T = ker
In each case, find a basis B = {e1,..., er, er+1,..., en} of V such that {er+1,..., en) is a basis of ker T, and verily Theorem 5. (a) T: R3 → R4; T(x, y, z) = (x -y + 2z, x + y - z, 2x + z, 2y -
In each case either prove the statement or give an example in which it is false. Throughout, let T: V → W be a linear transformation where V and W are finite dimensional. (a) If V = W, then ker T
Show that linear independence is preserved by one-to-one transformations and that spanning sets are preserved by onto transformations. More precisely, if T: V → W is a linear transformation, show
Given {v1,..., vn) in a vector space V, define T: Rn → V by T(r1,..., rn) = r1v1 + ∙ ∙ ∙ + rnvn. Show that T is linear, and that: (a) T is one-to-one if and only if (v1,..., vn] is
Verify that each of the following is an isomorphism (Theorem 3 is useful). (a) T: R3 → R3; T(x, y, z) = (x, x + y, x + y + z) (b) T: Mmn → Mmn; T(X) = UXV, U and V any vector space (c) T: V →
If S and T are both onto, show that ST is onto.
Let V -T→ U -s→ W be linear transformations. If ST is onto, show that S is onto and that dim W < dim U
Let T: V → V be a linear transformation. Show that T2 = 1 y if and only if T is invertible and T = T-l.
Let T: V → W be a linear transformation, and let {e1..., er, er+1,..., en) be any basis of V such that {er+1,..., en} is a basis of ker T. Show that im T = span{e1 ,..., er}.
Let V be the vector space of Exercise 4 §6.1. Find an isomorphism T: V → R2.
Let V consist of all sequences [x0, x1, x2,...) of numbers, and define vector operations [x0. x1,.....) + [y0, y1,.....) = [x0 + y0, x1 + y1,.....) r[x0, x1...) = [rx0 > rx1...) Define T: V → U and
Define T: Pn → Pn by T(p) = p(x) + xp'(x) for all p in Pn. Show that ker T = {0} and conclude that T is an isomorphism. [Write p(x) = a0 + a1x + ∙ ∙ ∙ ∙ + anxn and compare coefficients if
Let T: V → W be a linear transformation, where V and W are finite dimehsional. Show that T is onto if and only if there exists a linear transformation S: W → V with TS = 1w A [Let {e1,..., er
Let S and T be linear transformations V → W, where dim V = n and dim W = m. Show that im S = im T if and only if T = SR for some isomorphism R : V → V. [Show that dim(ker S) = dim(ker T) and
If T: V → U is a linear transformation where dim V - n, show that TST = T for some isomorphism S: V → V. [Let {e1,..., er, er+1,...,en] be as in Theorem 5 §7.2. Extend {T(e1),..., T(er)} to a
In each case, compute the action of ST and TS, and show that ST ‰ TS.(a) S: R3 †’ R3 with S(x, y, z) = (x, 0, z);T: R3 †’ R3 with T(x, y, z) = (x + y, 0, y + z)
In each case, show that the linear transformation T satisfies T2 = T.(a) T: R2 †’ R2; T(x, y) = (x + y, 0)
Determine whether each of the following transformations T has an inverse and, if so, determine the action of T-1.(a) T: R4 †’ R4;T(x, y, z, t) = (x + y, y + z, z + t, t + x)(b)(c) T: P,
In each case, show that T is self-inverse: T-l = T.(a) T: R2 †’ R2; T(x, y) = (ky - x, y), k any fixed number
In each case, show that T6 = 1R4 and so determine T-l. (a) T: R4 → R4; T(x, y, z, w) = (-y, x - y, z, -w)
In each case, show that T is an isomorphism by defining T-1 explicitly. (a) T: Mnn → Mn is given by T(A) = UA where U is invertible in Mnn.
Find a basis for the space V of sequences [xn) satisfying the following recurrences, and use it to find the sequence satisfying x0 = 1, x1, = 2, x2 = 1. xn+3 = -6xn + 7xn+1
In each case, find a basis for the space V of all sequences [xn) satisfying the recurrence, and use it to find xn if x0 = 1, x1 = -1, and x2 = 1. (a) xn+3 = -2xn + xn+1 + 2xn+2 (b) xn+3 = -6xn +
Find a basis for the space V of sequences (xn) satisfying each of the following recurrences. xn+2 = -abxn + (a + b)xn+l, (a ≠ b)
In the following case, find a basis of V. (a) V = {[|xn) | xn+4 = -xn+2 -2xn+3 for n > 0}
Find a basis for the space V of all sequences (xn) satisfying xn+2 = -xn. by the expansion theorem (Theorem 6 §5.3). Thus T is linear because (X + Y) • Ei = X • Ei + Y • Ei and (rX) • Ei =
In each case, use the Gram-Schmidt algorithm to convert the given basis B of V into an orthogonal basis. (a) V = R2, B = {[2 1], [1 2]} (b) V = R3, B = {[0 1 1], [1 1 1], [1 -2 2]}
If U is a subspace of Rn, show that projU(X) = X for all X in U.
If U is a subspace of Rn, show how to find an n × n matrix A such that U = [X|AX = 0}.
Think of Rn as consisting of rows. If A is m × n and AAT is invertible, show that E - AT(AAT)-1 A is a projection matrix.
In each case, write X as the sum of a vector in U and a vector in U⊥. (a) X = [2 1 6], U = span{[3 -1 2], [2 0 -3]} (b) X = [2 0 1 6], U = span{[l 1 1 1], [ 1 1 -1 -1], [1 -1 1 -1]} (c) X = [a b c
Let X = [l -2 1 6] in R4, and let U = span{[2 1 3 -4], [1 2 0 1]}. (a) Compute projU(X). (b) Use the basis in part (b) to compute ProjU(X).
In each case, use the Gram-Schmidt algorithm to find an orthogonal basis of the subspace U, and find the vector in U closest to X. (a) U = span{[l -1 0], [-1 0 1]}; X = [2 1 0] (b) U = span{[l -1 0
Let U = span{V1, V2,..., Vk), Vi in Rn, and let A be the k × n matrix with the Vi as rows. Use part (a) to find U⊥ if U= span{[l -1 2 1], [1 0 -1 1]}.
Let U be a subspace of Rn and let X be a vector in Rn. Using Exercise 7, or otherwise, show that X is in U if and only if X = projU(X).
Normalize the rows to make each of the following matrices orthogonal.(a)(b) (c) (d)
In each case find new variables y1 and y2 that diagonalize the quadratic form q. (a) q = x21 + 6x1x2 + x22 (b) q = x21 + 4x1x2 - 2x2
Show that the following are equivalent for a symmetric matrix A. (a) A is orthogonal. (b) A2 = I. (c) All eigenvalues of A are ±1. [For (b) if and only if (c), use Theorem 2.]
Assume that A and B are orthogonally similar (Exercise 12). Show that A and B are orthogonally similar.
Prove the converse of Theorem 3: If (AX) • Y = X • (AV) for all n-columns X and Y, then A is symmetric.
Let P be an orthogonal matrix. (a) Give 2 × 2 examples of P such that det P = 1 and det P = -1. (b) If P is n × n and det P ≠ (-1)n, show that I - P has no inverse.
If P is a triangular orthogonal matrix, show that P is diagonal and that all diagonal entries are 1 or -1.
(a) Let A be an m × n matrix. Show that the following are equivalent. (i) A has orthogonal rows. (ii) A can be factored as A = DP, where D is invertible and diagonal and P has orthonormal
Let A be a skew-symmetric matrix; that is, AT = -A. Assume that A is an n × n matrix. Show that P = (I - A)(I + A)-1 is orthogonal.
For each matrix A, find an orthogonal matrix P such that P-1AP is diagonal.(a)(b) (c)
Consider below value where one of a, c ‰ 0. Show that cA(x) - x(x - k)(x + k), where k = ˆša2 + c2 and find an orthogonal matrix P such that P-1AP is diagonal.
Find the Cholesky decomposition of each of the following matrices.(a)(b)
If A is positive definite, show that A = C2 where C is positive definite.
Show that a matrix A is positive definite if and only if A is symmetric and admits a factorization A = LDU as in (a).

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