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Linear Algebra with Applications 7th edition Steven J. Leon - Solutions

Let A be an m × n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.
Let x1,..., xk be linearly independent vectors in Rn, and let A be a nonsingular n × n matrix. Define yi = Axi for i = 1,....,k. Show that y1,....,yk are linearly independent.
Let {v1,..., vn} be a spanning set for the vector space V, and let v be any other vector in V. Show that v, v1, ..., vn are linearly dependent.
Let v1, v2,...,vn be linearly independent vectors in a vector space V. Show that v2...., vn cannot span V.
Let x1, x2,..., xk be linearly independent vectors in a vector space V. (a) If we add a vector xk+1 to the collection, will we still have a linearly independent collection of vectors? Explain. (b) If we delete a vector, say xk, from the collection, will we still have a linearly independent
For each of the following, show that the given vectors are linearly independent in C[0, 1]. (a) cos πx, sin πx (b) x3/2, x5/2 (c) 1, ex + e-x, ex - e-x (d) ex, e-x, e2x
Determine whether the vectors cosx, 1, sin2(x/2) are linearly independent in C[- π, π].
The vectorsspan R3. Pare down the set {x1, x2, x3, x4, x5} to form a basis for R3.
In R4, let U be the subspace of all vectors of the form (u1, u2, 0, 0)T, and let V the subspace of all vectors of the form (0, v2, v3, 0)T. What are the dimensions of U, V, U ∩ V, U + V? Find a basis for each of these four subspaces. (See Exercises 18 and 20 of Section 2.)
Is it possible to find a pair of two-dimensional subspaces U and V of R3 such that U ∩ V = {0}? Prove your answer. Give a geometrical interpretation of your conclusion.
Show that if U and V are subspaces of Rn and U ∩ V = {0}, then dim (U + V) = dim U + dim V
Given the vectors(a) Show that x1 and x2 form a basis for R2. (b) Why must x1, x2, x3 be linearly dependent?
Given(a) Show that x1, x2, x3 are linearly dependent. (b) Show that x1 and x2 are linearly independent.
Given x,=(1, 1, 1)T and x2 = (3, - 1, 4)T: (a) Do x1 and x2 span R3? Explain. (b) Let x3 be a third vector in R3 and set X = (x1, x2, x3). What condition(s) would X have to satisfy in order for x1, x2, x3 to form a basis for R3? (c) Find a third vector x3 that will extend the set {x1, x2} to a
Let a1 and a2 be linearly independent vectors in R3, and let x be a vector in R2. (a) Describe geometrically Span(a1, a2). (b) If A = (a1, a2) and b = Ax, then what is the dimension of Span(a1, a2, b)? Explain.
Let E = [u1, un] and F = [v1,..., vn] be two ordered bases for Rn and set U = (u1,..., un), V = (v1,...,vn) Show that the transition matrix from E to F can be determined by calculating the reduced row echelon form of (V|U).
For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space.(a)(b) (c)
Let A be a 4 Ã— 4 matrix with reduced row echelon form given byIf determine a3 and a4.
Let A be a 5 × 8 matrix with rank equal to 5 and let b be any vector inR5. Explain why the system Ax = b must have infinitely many solutions.
Let A be a 4 × 5 matrix. If a1, a2, a4 are linearly independent and a3 = a1 + 2a2 a5 = 2a1 - a2 + 3a4 determine the reduced row echelon form of A.
Let A be a 5 × 3 matrix of rank 3 and let {x1,x2,x3} be a basis for R3. (a) Show that N(A) = {0}. (b) Show that if y, = Ax1, y2 = Ax2, and y3 = Ax3, then y1, y2, y3 are linearly independent. (c) Do the vectors y1, y2, y3 from part (b) form a basis for R5? Explain.
Let A be an m × n matrix with rank equal to n. Show that if x ≠ 0 and y = Ax, then y ≠ 0.
Prove that a linear system Ax = b is consistent if and only if the rank of (A | b) equals the rank of A.
Let A be an m × n matrix. (a) If B is a nonsingular m × in matrix, show that BA and A have the same nullspace and hence the same rank. (b) If C is a nonsingular n × n matrix, show that AC and A have the same rank.
Prove Corollary 3.6.4.
Show that if A and B are n × n matrices and N(A - B) = Rn, then A = B.
Let A and B be n × n matrices. (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. (b) Show that if AB = O, then the sum of the ranks of A and B cannot exceed n.
Let A ∊ Rm×n, b ∊ Rm, and let x0 be a particular solution to the system Ax = b. Prove the following. A vector y in Rn will be a solution to Ax = b if and only if y = x0 + z, where z ∊ N(A). If N(A) = {0}, then the solution x0 is unique.
Let x and y be nonzero vectors in Rm and Rn, respectively, and let A = xyT. (a) Show that {x} is a basis for the column space of A and that {yT} is a basis for the row space of A. (b) What is the dimension of N(A)?
Let A ∊ Rm×n, B ∊ Rn×r and C = AB. Show that (a) The column space of C is a subspace of the column space of A. (b) The row space of C is a subspace of the row space of B. (c) rank(C) ≤ min{rank(A), rank(B)}.
Let A ∊ Rm×n, B ∊ Rn×r, and C = AB. Show that (a) If A and B both have linearly independent column vectors, then the column vectors of C will also be linearly independent. (b) If A and B both have linearly independent row vectors, then the row vectors of C will also be linearly independent.
Let A ∊ Rm×n, B ∊ Rn×r, and C = AB. Show that (a) If the column vectors of B are linearly dependent, then the column vectors of C must be linearly dependent. (b) If the row vectors of A are linearly dependent, then the row vectors of C are linearly dependent.
An m × n matrix A is said to have a right inverse if there exists an n x m matrix C such that AC = Im. A is said to have a left inverse if there exists an n × m matrix D such that DA = In. (a) If A has a right inverse, show that the column vectors of A span Rm. (b) Is it possible for an m × n
Show that a matrix B has a left inverse if and only if BT has a right inverse.
Let B be an n × m matrix whose columns are linearly independent. Show that B has a left inverse.
Prove that if a matrix B has a left inverse, the columns of B are linearly independent.
GivenWhich column vectors of A correspond to the lead variables of U? These column vectors form a basis for the column space of A. Write each of the remaining column vectors of A as a linear combination of these basis vectors.
If a matrix U is in row echelon form, show that the nonzero row vectors of U form a basis for the row space of U.
How many solutions will the linear system Ax = b have if b is in the column space of A and the column vectors of A are linearly dependent? Explain.
Let A be an m × n matrix with m > n. Let b ∊ Rm and suppose that N(A) = {0}. (a) What can you conclude about the column vectors of A? Are they linearly independent? Do they span Rm? Explain. (b) How many solutions will the system Ax = b have if b is not in the column space of A? How many
Let A and B be row equivalent matrices. (a) Show that the dimension of the column space of A equals the dimension of the column space of B. (b) Are the column spaces of the two matrices necessarily the same? Justify your answer.
In this exercise we consider how to generate matrices with specified ranks using MATLAB. (a) In general, if A is an m × n matrix with rank r, then r ≤ min(m, n). Why? Explain. If the entries of A are random numbers, we would expect that r = min(m, n). Why? Explain. Check this out in MATLAB by
Set B = round(K) * rand(8, 4)), X = round(10 * rand(4, 3)), C = B * X, and A = [ B C ].(a) How are the column spaces of B and C related? (See Exercise 22 in Section 6.) What would you expect the rank of A to be? Explain. Use MATLAB to check your answer.(b) Which column vectors of A should form a
To see why the rank 1 update method works, use MATLAB to compute and compare Cy and b + cu Prove that if all computations had been carried out in exact arithmetic these two vectors would be equal. Also compute Cz and (l + d)u Prove that if all computations had been carried out in exact arithmetic
If S is a subspace of a vector space V, then S is a vector space.
R2 is a subspace of R3.
It is possible to find a pair of two-dimensional subspaces S and T of R3 such that S ∩ T = {0}.
If x1, x2,...., xn span a vector space V, then they are linearly independent.
If x1, x2,...., xk are vectors in a vector space V and Span(x1, x2,...., xk) = Span(x1, x2,...., xk-1) then x1, x2,...., xk are linearly dependent.
If A is an m × n matrix, then A and AT have the same rank.
Let A be a 6 × 5 matrix with linearly independent column vectors a1, a2, a3 and whose remaining column vectors satisfy a4 = a1 + 3a2 + a3, a5 = 2a1 - a3 (a) What is the dimension of N(A)? Explain. (b) Determine the reduced row echelon form of A.
Let [u1, u2] and [v1, v2] be ordered bases for R2. Where(a) Determine the transition matrix corresponding to change of basis from the standard basis [e1,e2] to the ordered basis [111,112]. Use this transition matrix to find the coordinates of with respect to [u1, u2]. (b) Determine the transition
For each of the following determine whether the given set is a subspace of R2. Prove your answers.(a)(b)
Given(a) Find a basis for N(A) (the null space of A). What is the dimension of N(A)? (b) Find a basis for the column space of A. What is the rank of A?
How do the dimensions of the nullspace and column space of a matrix relate to the number of lead and free variables in the reduced row echelon form of the matrix? Explain.
Answer the following questions and in each case give geometric explanations of your answers. (a) Is it possible to have a pair of one dimensional subspaces U1 and U2 of R3 such that U1 ∩ U2 = {0}? (b) Is it possible to have a pair of two dimensional subspaces V1 and V2 of R3 such that V1 ∩ V2 =
Let S be the set of all symmetric 2 × 2 matrices with real entries. (a) Show that S is a subspace of R2×2. (b) Find a basis for S.
Let A be a 6 × 4 matrix of rank 4. (a) What is the dimension of N(A)? What is the dimension of the column space of A? (b) Do the column vectors of A span R6 Are the column vectors of A linearly independent? Explain your answers. (c) How many solutions will the linear system Ax = b have if b is in
Given the vectors(a) Are x1, x2, x3 x4 linearly independent in R3? Explain. (b) Do x1 and x2 span R3? Explain. (c) Do x1, x2 and x3 span R3? Are they linearly independent? Do they form a basis for R3? Explain. (d) Do x1, x2, and x4 span R3? Are they linearly independent? Do they form a basis for
Let x1, x2, x3 be linearly independent vectors in R4 and let A be a nonsingular 4 × 4 matrix. Prove that if y1 = Ax1, y2 = Ax2, y3 = Ax3 then y1, y2, y3 are linearly independent.
For each f ˆˆ C[0. 1] define L,(f) = F, whereShow that L is a linear operator on C[0, 1] and then find L(ex) and L(x2).
If L is a linear transformation from V to IV, use mathematical induction to prove that L(a1v1 + a2v2 +...... + anvn) = a1L(V2) + a2L(v2) +.......+ anL(vn)
Let (v1,...,vn) be a basis for a vector space V, and let L1 and L2 be two linear transformations mapping V into a vector space W. Show that if L1(v1) = L2(v1) for each i = I,... ,11 then L1 = L2 [i.e., show that L1(v) = L2(v) for all v ∈ V).
Let L be a linear operator on R1 and let a = L(l). Show that L(x) = ax for all .v ∈ R1
Let L be a linear operator on a vector space V. Define Ln, n ≥ 1, recursively by L1 = L Lk+ l(v) = L(Lk(V)) for all v ∈ V Show that L" is a linear operator on V for each n > 1.
Let L1 : U → V and L2 : V → W be linear transformations, and let L = L2 ° L1 be the mapping defined by L(u) = L2(L,(u)) for each u ∈ U. Show that L is a linear transformation mapping U into W
Determine the kernel and range of each of the following linear operators on P3.(a) L(p(x)) = xp'(x) (b) L(p(x)) = p(x) - p'(x) (c) L(p(x)) = p(0)x + p(1)
Let L be the linear operator on R2 defined by L(x) = (x1) cos α - x2 sin α, x1 sin α + x2 cos α)T Express x1, x2, and L(x) in terms of polar coordinates. Describe geometrically the effect of the linear transformation.
Let L: V → IV be a linear transformation, and let T be a subspace of W. The inverse image of T denoted L-l(T), is defined by L-1(T) = {v ∈ V\L(v) ∈ T} Show that L-1(T) is a subspace of V
A linear transformation L: V → W is said to be one-to-one if L(V1) = L(v2) implies that v1 = v2 (i.e., no two distinct vectors v1, v2 in V get mapped into the same vector w ∈ W). Show that L is one-to-one if and only if ker(L) = {0v}.
A linear transformation L: V → W is said to map V onto W if L(V) = W. Show that the linear transformation L defined byL(x) = (x1.x1 +x2,x1 +.x2 + x3)Tmaps R3 onto R3.
Let A be a 2 × 2 matrix, and let LA be the linear operator defined byLA(x) = AxShow that(a) LA maps R2 onto the column space of A.(b) If A is nonsingular, then LA maps R2 onto R2.
Let D be the differentiation operator on P3, and let 5 = [p ∈ P3| p(0) = 0) Show that (a) D maps P3, onto the subspace P2, but D: P3 → P2 is not one-to-one. (b) D: S → P3 is one-to-one but not onto.
Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a translation. Show that a translation is not a linear operator. Illustrate geometrically the effect of a translation.
Let L: R2 → R2 be a linear operator. If L((1,2)T) = (-2,3)T and L((l,-1)T) = (5, 2)T determine the value of L((7, 5)T).
Let C be a fixed n x n matrix. Determine whether the following are linear operators on R n × n (a) L(A)= CA + AC (b) L(A) = C2A (c) L(A) = A2C
Determine the matrix representation of each of the following composite transformations. (a) A yaw of 90°, followed by a pitch of 90° (b) A pitch of 90°, followed by a yaw of 90J (c) A pitch of 45°, followed by a roll of -90c (d) A roll of -90°, followed by a pitch of 45° (e) A yaw of 45°,
Let Y, P, and R be the yaw, pitch, and roll matrices given in equations (1), (2) and (3) and let Q = YPR.(a) Show that Y, P, and R all have determinants equal to 1.(b) The matrix Y represents a yaw with angle u. The inverse transformation should be a yaw with angle - u. Show that the matrix
Let L be a linear operator on Rn. Suppose that L(x) =0 for some x ≠ 0. Let A be the matrix representing L with respect to the standard basis [e1, e2,..., en]. Show that A is singular
Let L be a linear operator on a vector space V. Let A be the matrix representing L with respect to the ordered basis [v1,...,vn] of V [i.e., L(j) =Show that A'" is the matrix representing L'" with respect to [v;, ..., v,,].
Suppose that L1: V → W and L2: Z are linear transformations and E, F, and G are ordered bases for V, W, and Z, respectively. Show that, if A represents L1 relative to E and F and 6 represents L2 relative to F and G, then the matrix C = BA represents L2 ° Ll : V → Z relative to F and G. [Hint
Let V, W be vector spaces with ordered bases F and F, respectively. If L : V → W is a linear transformation and A is the matrix representing L relative to F and F, show that(a)v ∈ ker(L) if and only if [v]E ∈ N(A).(b)w ∈ L(V) if and only if [w]f is in the column space of A
Letand let I be the identity operator on R3. Find the coordinates of I(e1), I(e2), and I(e3) with respect to [y1, y2, y3].
Let A and B be n x n matrices. Show that if A is similar to B then there exist n x n matrices S and 7, with S nonsingular, such that A = ST and B = TS
Show that if A and B are similar matrices then det(A) = det( B)
Let A and B be similar niatrices. Show that(a) AT and BT are similar.(b) Ak and Bk are similar for each positive integer k.
Show that if A is similar to 1? and A is nonsingular. then B must also be nonsngular and A-1and B-1 are similar.
Let A and B be similar matrices and let A be any scalar. Show that(a) A - λ1 and B - λl are similar. (b) det(A - λl)=del(B - λl).
The trace of an n × n matrix A. denoted tr(A). is the sum of its diagonal entries: that is. tr(A) = α11 + α22 +......+(αnn) Show that (a) tr (AB) = tr(BA) (b) If A is similar to B, then tr(A) = tr(B).
Prove that if A is similar to B and B is similar to C, then A is similar to C.
Suppose that A = SAS-1 where Λ is a diagonal matrix with diagonal elements λ1, λ2, . . . , λn.(a) Show that ASi = λiSi i = 1,..., n.(b) Show that if x = a1S1 + a2s2 + αns2 +.....+ then Akx = α1λk1s1 + α2λk2s2 +......+ αnλknsn(c) Suppose that |λi| < 1 for I = 1,....n. What happens to
Suppose that A = ST, where S is nonsingular. Let B = TS. Show that B is similar to A
Set A = triu(ones(5)) * trill one s(S). If L dcnoc, the linear operator defined L(x) = Ax for all x in Rn. then A is the matrix representing L. with respect to the standard basis for R5. Construct a 5s x 5 Inatro. U by settingU = hankel(ones(5. 1) 1: 5).Use the MATLAB function rank to verify that
Setx = [0 : 4,4,-4. 1. 1]' and y = ones(9, 1)(a) Use the MATLAB function norm to compute the values of ||x||, ||y||, ||x + y|| and to verify that the triangle inequality holds. Use MATLAB also to verify that the parallelogram law||x + y||2 + ||x - y||2 = 2(||x||2 + ||y||2)is satisfied.(b)
The vector spaces N(A), R(A), N(AT), R(AT) are the four fundamental subspaces associated with a matrix A. We can use MATLAB to construct orthonormal bases for each of the fundamental subspaces associated with a given matrix. We can then construct projection matrices corresponding to each
If x and y are nonzero vectors in Rn and the vector projection of x onto y is equal to the vector projection of y onto x, then x and y are linearly dependent.
If {u1, u2,..., uk} is an orthonormal set of vectors in Rn and U = (ul, u2, uk) then UUT = In (the n × n identity matrix).
If x and y are unit vectors in Rn and |xTy| = 1, then x and y are linearly independent.
If U, V, and W are subspaces of R3 and U ⊥ V and V ⊥W, then U ⊥W.
It is possible to find a nonzero vector y in the column space of A such that ATy = 0.

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