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Questions and Answers of Linear Algebra

In each case solve the problem by finding the matrix of the operator. (a) Find the projection of = [0 1 -3]T on the plane with equation 2x - y + 4z = 0. (b) Find the reflection of = [0 1 -3]T in
Find the rotation of = [1 0 3]T about the z axis through θ = π/6.
Find the matrix of the rotation about the y axis through the angle θ (from the positive x axis to the positive z axis).
Let L be the line through the origin in R2 with direction vector = [a b]T ‰ 0.If PL denotes projection on L, show that PL has matrix
Consider the letter 4 described in Figure 4.38. Find the data matrix for the letter obtained by:Rotating the letter through Ï€/4 about the point
Find the reflection of the point P in the line y = 1 + 2x in R2 if: P = P(1, 4)
A wind is blowing from the south at 75 knots, and an airplane flies heading east at 100 knots. Find the resulting velocity of the airplane.
A rescue boat has a top speed of 13 knots. The captain wants to go due east as fast as possible in water with a current of 5 knots due south. Find the velocity vector = (x, y) that she must
In each case determine whether U is a subspace of R3. Support your answer. (a) U = {[1 s t]T | s and t in R}. (b) U = {[0 s t]T | s and t in R}. (c) U = {[r s t]T | r, s, and t in R, - r + 3s + 2t =
If a1, a2,..., ak are nonzero scalars, show that span {a1X1, a2X2,..., akXk} = span{X1 ,X2,...,Xk) for any vectors X, in Rn.
Suppose that U = span {X1, X2,Xk} where each Xi is in Rn. If A is an m × n matrix and AXi = 0 for each i, show that AY=0 for every vector Y in U.
In each case either show that the statement is true or give an example showing that it is false. (a) If U≠ Rn is a subspace of Rn and X+ Y is in U, then X and F are both in U. (b) If U is a
In each case determine if X lies in U = span{Y, Z}. If X is in U, write it as a linear combination of Y and Z; if X is not in U, show why not. (a) X = [2 -1 0 1]T, Y=[l 0 0 1]T, and Z=[0 1 0 1]T. (b)
Let U be a nonempty subset of Rn. Show that U is a subspace if and only if S2 and S3 hold.
Let U and W be subspaces of Rn. Define their intersection U ∩ W and their sum U + W as follows: U ∩ W={X in Rn | X belongs to both U and W}. U+ W= {X in Rn | X is a sum of a vector in U and a
In each case determine if the given vectors span R4. Support your answer. (a) {[1 1 1 1]T, [0 1 1 1]T, [0 0 1 l]T, [0 0 0 l]7}. (b) {[1 3 -5 0]T, [-2 1 0 0]T, [0 2 1 -l]T, [1 -4 5 0]T}.
Which of the following subsets are independent? Support your answer. (a) {[1 -1 O]T, [3 2 -1]T, [3 5 -2]T) in R3. (b) [(1 1 1]T, [1 -1 l]T, [0 0 l]T} in R3. (c) {[1 -1 1 -1]T, [2 0 1 0]T, [0 -2 1
If {X1, X2, X3, X4, X5, X6} is an independent set of vectors, show that the subset {X2, X3, X5} is also independent.
If {X1 X2, X3,..., Xk} is independent, show that {X1, X1 + X2, X1, + X2 + X3,..., X1 + X2 + ∙ ∙ ∙ + Xk} is also independent.
Suppose that {X, Y) is a basis of R2, and let(a) If A is invertible, show that {aX+bY, cX+ dY} is a basis of R2. (b) If [aX + bY, cX+dY) is a basis of R2, show that A is invertible.
Let A denote an m × n matrix. (a) Show that null A = null(UA) for every invertible m × m matrix U. (b) Show that dim(null A) = dim(null (AV)) for every invertible n × n matrix V.
Let {X, Y, Z, IV) be an independent set in Rn. Which of the following sets is independent? Support your answer. (a) {X - Y, Y-Z,Z-X] (b) {X + Y, Y+Z,Z + X) (c) {X - Y, Y-Z, Z- W, W - X) (d) {X + Y,
Let U and W denote subspaces of Rn, and assume that U ⊆ W. If dim W = 1, show that either U = {0} or U = W.
Find a basis and calculate the dimension of the following subspaces of R4. (a) span{[l -1 2 O], [2 3 0 3]T, [1 9 -6 6]T}. (b) span{[2 1 0 -1]T, [-1 1 1 1]T, [2 7 4 5]T}. (c) span{[-l 2 1 0]T, [2 0 3
Find a basis and calculate the dimension of the following subspaces of R4. (a) U-{[a a + b a-b b]T|a and b in R}. (b) U={[a + b a-b b a]T | a and b in R}. (c) U={[a b c + a c]T |a, b, and c in
Suppose that {X, Y, Z, W) is a basis of R4. Show-that: (a) [X + aW, Y, Z, W) is also a basis of R4 for any choice of the scalar a. (b) {X+ W, Y+ W, Z + IV, W] is also a basis of R4. (c) {X, X+ Y, X+
Use Theorem 3 to determine if the following sets of vectors are a basis of the indicated space. (a) {[3 -1]T, [2 2]T}inR2. (b) {[1 1 -1]T, [1 -1 1]T, [0 0 l]T}inR3. (c) {[-1 1 -l]T, [l -1 2]T, [0 0
In each case show that the statement is true or give an example showing that it is false. (a) If {X, Y} is independent, then {X, Y, X + Y} is independent. (b) If {X, Y, Z} is independent, then {Y, Z}
Obtain orthonormal bases of R3 by normalizing the following. (a) {[1 -1 2]T, [0 2 1]T, [5 1 -2]T) (b) {[1 1 1]T, [4 1 -5]T,[2 -3 1]T}
Use the Cauchy inequality to prove that" (a) ( r1 +r2 + . . . + rn )n ≤ n (r21 + r22 + . . . + r2n for all ri in R and all n ≥ 1. (b) r1r2 + r1r3 + r2r3 ≤ r21 + r22 + r23 for all r1, r2,and r3
(a) Show that X and Y are orthogonal in Rn if and only if || X + Y || = ||X - Y||. (b) Show that X + Y and X - Y are orthogonal in Rn if and only if || X || = || Y ||.
If A is n × n, show that every eigenvalue of ATA is nonnegative. [Compute ||AX||2 where X is an eigenvector.]
In each case, show that the set of vectors is orthogonal in R4. (a) {[1 -1 2 5]T, [4 1 1 -1]T, [-7 28 5 5]T] (b) {[2 -1 4 5]T, [0 -1 1 -l]T,[0 3 2 -1]T}
In each case show that B is an orthogonal basis of R3 and use Theorem 6 to expand X = [ a b c]T as a linear combination of the basis vectors. (a) B = {[1 -1 3]T,[-2 1 1]T, [4 7 1]T} (b) B = {[1 0
In each case, write X as a linear combination of the orthogonal basis of the subspace U. (a) X = [13 -20 15]T; U = span{[l -2 3]T, [-1 1 1]T} (b) X = [14 1 -8 5]T ; U = span{[2 -1 0 3]T, [2 1 -2 -]T}
In each case, find all [a b c d]T m R4 such that the given set is orthogonal. (a) {[1 2 1 O]T, [1 -1 1 3]T, [2 -1 0 -1], [a b c d]T} (b) {[1 0 - 1]T, [2 1 1]T,[l -3 1 0]T, [a b c d}T}
If ||X|| = 3, ||F|| = 1, and X∙F= -2, compute: (a) || 3X - 5F|| (b) ||2X + 7Y|| (c) (3X-F)∙(2F-X) (d) (X-2F)∙(3X + 5F)
In each case either show that the statement is true or give an example showing that it is false. (a) Every independent set in Rn is orthogonal. (b) If {X, Y) is an orthogonal set in Rn, then {X, X +
If A is an m × n matrix with orthonormal columns, show that ATA = In.
Let A be an n × n matrix. (a) Show that A2 = 0 if and only if col A ⊆ null A. (b) Conclude that if A2 = 0, then rank A ≤ n/2. (c) Find a matrix A for which col A = nullA.
If A is an m × n matrix, show that col A = {AX| X in Rn}
(a) Show that AX = B has a solution if and only if rank A = rank[A B]. (b) lf [AX] = B has no solution, show that rank[A B] = 1 + rank A
In each case find a basis of the null space of A. Then compute rank A and verify (1) of Theorem 2.(a)(b)
Let A = CR where C ≠ 0 is a column in Rm and R ≠ 0 is a row in Rn. (a) Show that col A = span {C} and row A = span {R}. (b) Find dim(null A). (c) Show that null A = null R.
Let A be m × n with columns C1, C2, . . . , Cn. (a) If {C1, . . ., Cn} is independent, show null A = 0. (b) If null A = 0, show that {C1, . . .,Cn) is independent.
By computing the trace, determinant, and rank, show that A and B are not similar in each case.(a)(b) (c) (d) (e) (f)
Let A be a diagonalizable n × n matrix with eigenvalues λ1, λ2, . . ., λn(including multiplicities). Show that: (a) det A = λ1λ2 . . . λn (b) trA = λ1 + λ2 + . . . + λn
Let p be an invertible n × n matrix. If A is any n × n matrix, write Tp(A) = P-1AP. Verify that: (a) Tp (I) = I (b)TP(AB) = TP(A)Tp(B) (c) Tp(A + B) = Tp(A) + Tp(B) (d) Tp(rA) = rTp(A) (e) Tp(Ak) =
(a) Show that two diagonalizable matrices are similar if and only if they have the same eigenvalues with the same multiplicities.(b) If A is diagonalizable, show that A ~ AT.(c) Show that A ~ AT if
(a) Show that x3 - (a2 + b2 + c2)x - 2 abc has real roots by considering A.(b) Show that a2 + b2 + c2 ‰¥ ab + ac + be by considering B.Let
Refer to Section 3.4 on linear recurrences. Assume that the sequence x0, .xn x2,... satisfiesThen show that:xn + k = r0xn + r1xn+1 + ˆ™ ˆ™ ˆ™ + rk-1xn+k-1for all n
If A ~ B, show that: (a) AT ~ BT (b) A-1 ~ B-1 (c) rA~ rB for r in R (d) An ~ Bn for n ≥ 1
In each case, decide whether the matrix A is diagonalizable. If so, find P such that P-1 AP is diagonal.(a)(b) (c) (d)
If A ~ B and A has any of the following properties, show that B has the same property. (a) Idempotent, that is A2 = A. (b) Nilpotent, that is Ak= 0 for some k ≥. (c) Invertible
Let A denote an n Ã— n upper triangular matrix.(a) If all the main diagonal entries of A are distinct, show that A is diagonalizable.(b) if all the main diagonal entries of A are equal,
Find the best approximation to a solution of each of the following systems of equations. (a) x + y - z = 5 2x - y + 6z = 1 3x + 2y - z = 6 -x + 4y + z = 0 (b) 3x + y + z = 6 2x + 3y - z = 1 2x -
(a) Use m = 0 in Theorem 3 to show that the best-fitting horizontal line y = a0 through the data points (x1,y1),..., (xn,yn) is y = 1/n(y1+y2 + ∙ ∙ ∙ +yn), the average of the y coordinates. (b)
Given the situation in Theorem 4, write f(x) = r0p0 (x) + r1p1(x) + ∙ ∙ ∙ + rmpm (x) Suppose that f(x) has at most k roots for any choice of the coefficients r0,r1,..., rm, not all zero. (a)
Find the least squares approximating line y = a0 + a1x for each of the following sets of data points. (a) (1, 1), (3, 2), (4, 3), (6, 4) (b) (2, 4), (4, 3), (7, 2), (8,1) (c) (-1,-1), (0, 1), (1,2),
Find the least squares approximating quadratic y = a0 + a1x for each of the following sets of data points. (a) (0,1), (2, 2), (3, 3), (4, 5) (b) (-2, 1), (0, 0), (3,2), (4, 3)
Find a least squares approximating function of the form r0x + r1x2 + r22x for each of the following sets of data pairs. (a) (-1, 1),(0, 3),(1, 1),(2,0) (b) (0, 0,(1,1), (2, 5), (3,10)
Find the least squares approximating function of the form r0 + r1x2 + r2 sin πx/2 for each of the following sets of data pairs. (a) (0, 3), (1, 0), (1, -1), (-1, 2) (b) (-1, 1/2), (0, 1), (2, 5),
Newton's laws of motion imply that an object dropped from rest at a height of 100 metres will be at a height s = 100 -1/2gt2 metres t seconds later, where g is a constant called the acceleration due
The yield y of wheat in bushels per acre appears to be a linear function of the number of days x1 of sunshine, the number of inches x2 of rain, and the number of pounds x3 of fertilizer applied per
The following table gives the number of years of education and the annual income (in thousands) of 10 individuals. Find the means, the variances, and the correlation coefficient. (Again the data
Prove the data scaling formulas found on page 279: (a), (b) and (c).
In each case either show that the statement is true or give an example showing that it is false. Throughout, X, Y, Z, X1, X2,..., Xn denote vectors in Rn. (a) If U is a subspace of Rn and X+Y is in
Let V denote the set of ordered triples (x, y, z) and define addition in V in R3. For each of the following definitions of scalar multiplication, decide whether V is a vector space. (a) a(x, y, z) =
Show that the zero vector 0 is uniquely determined by the property in axiom A4.
Prove that (- a)v = - (ax) in Theorem 3 by first computing (- a)v + av. Then do it using (4) of Theorem 3 and axiom S4.
Let v, v1,..., vn denote vectors in a vector space V and let a, a1,..., an denote numbers. Use induction on n to prove each of the following. (a) a(a1 + a2 + ... + an)v = a1v + a2v + ... + anv
Prove the following for vectors u and v and scalars a and b. (a) If av = aw and a ≠ 0, then v = w.
Are the following sets vector spaces with the indicated operations? If not, why not? (a) The set V of all polynomials of degree ≥ 3, together with 0; operations of P. (b) The set {1, x, x2,...};
If V is the set of ordered pairs (x, y) of real numbers, show that it is a vector space if (x, y) + (x1, y1) = (x + x1, y + y1 + 1) and a(x, y) = (ax, ay + a - 1). What is the zero vector in V?
Find x and y (in terms of u and v) such that: 3x - 2y = u 4x - 5y = v
In each case show that the conditionau + bv + cw = 0 in V implies that a = b = c = 0.(a)(b) V = F[0, Ï€]; u = sin x, v = cos x, w = 1
Simplify the following. (a) 4(3u - v + w) - 2[(3u - 2v) - 3(v - w)] + 6(w - u - v)
Which of the following are subspaces of P3? Support your answer. (a) U = {f(x) | f(x) in P3, f(2) = 1} (b) U= {xg(x) + (1 - x)h(x) | g(x) and h(x) in P2} (c) U = {f(x) | f(x) in P3, degf(x) = 3}
Let u, v, and w denote vectors in a vector space V. Show that: span{u, v, w} = span{u - v, u + w, w}
Is it possible that {(1, 2, 0), (1, 1, 1)} can span the subspace U = {(a, b, 0) | a and b in R}?
Suppose V= span{v1, v2,..., vn}. If u = a1v1 + a2v2 + ... + anvn where the ai, are in R and a1 ≠ 0, show that V = span{u, v2,..., vn).
Which of the following are subspaces of Mi?? Support your answer.(a)(b) U = {A | A in M22, AB = 0}, B a fixed 2 Ã— 2 matrix (c) U = {A | A in M22, A is not invertible}
Let U be a subspace of a vector space V. If u and u + v are in U, show that v is in U.
Let U be a nonempty subset of a vector space V. Show that U is a subspace of V if and only if u1 + au2 lies in U for all u1 and u2 in U and all a in R.
Which of the following are subspaces of F[0, 1]? Support your answer. (a) U = {f | f(0) = 0} (b) U = {f | f(x) ≥ 0 for all x in [0, 1]} (c) U = {f | f(x + y) = f(x) + f(y) for all x and y in [0, 1]}
Let A be a vector in Rn (written as a column), and define U = {AX | A in Mmn}. Show that U = Rm if X ≠ 0.
Write each of the following as a linear combination of x + 1, x2 + x, and x2 + 2. (a) 2x2 - 3x + 1 (b) x
Determine whether v lies in span{u, w} in each case.(a) v = x; u = x2 + 1, w = x + 2(b)
Which of the following functions lie in span{cos2x, sin2x}? (Work in F[0, π].) (a) 1 (b) 1 + x
Show that P2 is panned by {1 + 2x2, 3x, 1 + x}.
Show that each of the following sets of vectors is independent.(a) {x2, x + 1, 1 - x - x2} in P2(b)
(a) Let V denote the set of all 2 × 2 matrices with equal column sums. Show that V is a subspace of M22, and compute dim V. (b) Repeat part (a) for 3 × 3 matrices. (c) Repeat part (a) for n × n
(a) Let V= {(x2 + x + 1)p(x) | p(x) in P2}. Show that V is a subspace of P4 and find dim V. (b) Repeat with V= {{x2 - x)p{x) | p(x) in P3}, a subset of P5. (c) Generalize.
In each case, either prove the assertion or give an example showing that it is false. (a) P2 has a basis of polynomials f(x) such that f(b) = 0. (b) Every basis of M22 contains a noninvertible
Show that every nonempty subset of an independent set of vectors is again independent.
Assume that {u, v} is independent in a vector space V. Write uÊ¹ = au + bv and vÊ¹ = cu + dv, where a, b, c, and d are numbers. Show that {uÊ¹, vÊ¹} is
Which of the following subsets of Fare independent? (a) V= P2; {x2 - x + 3, 2x2 + x + 5, x2 + 5x + 1} (b) V = M22; (c) V = F[0, 1];
Let {u, v, w, z] be independent. Which of the following are dependent? (a) {u + v, v + w, w + u} (b) {u + v, v + w, w + z, z + u}
If z is a complex number, show that {2, z2} is independent if and only if z is not real.
Show that part (a) fails if U is not invertible.

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