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linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

Write the difference equations as first order systems, xk+1 = Axk, for all k. Yk+4-2yk +3-3yk+2 +8yk+1-4yk = 0
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If there exists a linearly independent set {v1,.....,vp} in V, then dim V ≥ P.
Concern finite-dimensional vector spaces V and W and a linear transformation T: V → W.Let H be a nonzero subspace of V, and let T(H) be the set of images of vectors in H. Then T(H) is a subspace of
Concern finite-dimensional vector spaces V and W and a linear transformation T: V → W.Let H be a nonzero subspace of V, and suppose T is a one-to-one (linear) mapping of V into W. Prove that dim
Let f, g, and h be linearly independent functions defined for all real numbers, and construct three signals by sampling the values of the functions at the integers:Must the signals be linearly
Justify the following equality: dim Row A + nullity AT = m, the number of rows of A.
Justify the following equality: dim Row A + nullity A = n, the number of columns of A.
Describe an infinite linearly independent subset of the subspace W in Exercise 29. Does this establish that W is infinite dimensional? Justify your answer.Data from in Exercise 29LetA typical signal
Show that the space C(R) of all continuous functions defined on the real line is an infinite-dimensional space.
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If p ≥ 2 and dim V = p, then every set of p - 1 nonzero vectors is linearly independent.
Describe an infinite linearly independent subset of the subspace W in Exercise 30. Does this establish that W is infinite dimensional? Justify your.Data from in Exercise 30LetA typical signal in W
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If every set of p elements in V fails to span V, then dim V > p.
Is the following difference equation of order 3? Explain. Yk+3 + 5yk +2 +6yk+1 = 0
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If dim V = p, then there exists a spanning set of p + 1 vectors in V.
Let yk = k2 and Zk = 2k|k]. Are the signals {yk} and {zk} linearly independent? Evaluate the associated Casorati matrix C(k) for k = 0, k = -1, and k = -2, and discuss your results.
Write the difference equations as first order systems, xk+1 = Axk, for all k. - Yk + 3 Yk +2 + 1/6 Yk =
Show that the given signal is a solution of the difference equation. Then find the general solution of that difference equation. Yk = 22k; Yk+22k+1 + 2yk = 2 + 3k
Show that the given signal is a solution of the difference equation. Then find the general solution of that difference equation. Yk = 2k − 4; Yk+2 + ¾Yk+1 − Yk = 1 + 3k -
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If there exists a linearly dependent set {v1,...,vp} in V, then dim V ≤ P.
What is the order of the following difference equation? Explain your answer. Yk+3 + a1yk +2 + a2yk +1+A3 yk = 0
V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Justify each answer.(T/F) If there exists a set {v1,...,vp} that spans V, then dim V ≤ p.
Explain why the space P of all polynomials is an infinite-dimensional space.
If A is a 4 x 3 matrix, what is the largest possible dimension of the row space of A? If A is a 3 x 4 matrix, what is the largest possible dimension of the row space of A? Explain.
If the nullity of a 7 x 6 matrix A is 5, what are the dimensions of the column and row spaces of A?
If the nullity of a 5 x 6 matrix A is 4, what are the dimensions of the column and row spaces of A?
LetA typical signal in W looks likeShow that W is a subspace of S. W v = {{xa} \x₁ = {ve if k < 0 if k ≥ 0 where each rk can be any real number
Suppose a 5 x 6 matrix A has four pivot columns. What is nullity A? Is Col A = R4? Why or why not?
Suppose a 5 x 9 matrix A has four pivot columns. Is Col A = R5? Is Nul A = R4? Explain your answers.
Let B be the basis of P2 consisting of the first three Laguerre polynomials listed in Exercise 28, and let p(t) = 7 - 8t + 3t2. Find the coordinate vector of p relative to B.Data from in Exercises
If a 6 x 3 matrix A has rank 3, find nullity A, rank A, and rank AT.
If a 4 x 7 matrix A has rank 4, find nullity A, rank A, and rank AT.
Let H be an n-dimensional subspace of an n-dimensional vector space V. Show that H = V.
Let S be a subset of an n-dimensional vector space V, and suppose S contains fewer than n vectors. Explain why S cannot span V.
Let B be the basis of P3 consisting of the Hermite polynomials in Exercise 27, and let p(t) = 7 - 12t - 8t2 + 12t3. Find the coordinate vector of p relative to B.Data from in Exercise 27The first
LetA typical signal in W looks likeShow that W is a subspace of S. W = - fixed 1 xe = {0 {Xk} (0 if k is a multiple of 2 if k is not a multiple of 2 where each r, can be any real number. rk
Find a basis for the set of vectors in R2 on the line y = 5x.
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 1 0 3 -1 2 2 1 06
Determine ifis in Nul A, where W = 1 3 -4
Verify that det AB = (det A)(detB) for the matrices. 2 2 A-[33] [34] B = 3= -1 А
Determine ifis in Nul A, where W = in -3 2
Find the vector x determined by the given coordinate vector [x]B and the given basis B. 3 B-{]-[ ]-AND-[] B= = [x g 3
Find the vector x determined by the given coordinate vector [ x ]B and the given basis B. B = (HL) -5 7 -4 -8 6 9 ‚[X]µ ° 2 -3 0
Find an explicit description of Nul A by listing vectors that span the null space. A = 1 3 5 0 54 14 -2
Find the vector x determined by the given coordinate vector [ x ]B and the given basis B. B = 8 [][] = [] ₂[x] B
Find the vector x determined by the given coordinate vector [ x ]B and the given basis B. []-~CD)- L- B = 0 , [x]g
Find an explicit description of Nul A by listing vectors that span the null space. =[1 0 A = -6 4 02 0
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. 0 AAA 0 I
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. 2 1 HIGH -2 -3 2 -7 54
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. 1 0 -2 3 2 -4 -3 -5 1
Find the coordinate vector [x]B of x relative to the given basis B = {b1,.....,bn}. [2²- ] - [ ] = 4 [F] = 4 =
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. 0 HBO] 0 1 0
Justify your answer.Suppose F is a 5 x 5 matrix whose column space is not equal to R5. What can you say about Nul F?
Justify each answer. Assume that all matrices here are square.(T/F) If A3 = 0, then det A = 0.
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. -3 0 0 0 [HH] 0
Find an explicit description of Nul A by listing vectors that span the null space. 1 5 -4 -2 0 A 01 0 0 -3 10 0 0
Find an explicit description of Nul A by listing vectors that span the null space. 1 -2 A = 0 0 0 0 4 0 0 1-9 0001
Find bases for the null spaces of the matrices given. Refer to the remarks that follow Example 3.Data from in Example 3 0-3 0 1 -5 3-2 2 4 1-2
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. 1 -4 [F] 2 -5 -3 6
Find the coordinate vector [ x ]B of x relative to the given basis B = {b1,.....,bn}. b₁ » = [-2]· ► = [ 8 ]- x = [8] b₂ X -6
Find bases for the null spaces of the matrices given. Refer to the remarks that follow Example 3 in Section 4.2.Data from in Example 3 1 0 -5 1 4 -2 1 6 -2 -2 0 2 -8 19
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. -2 [][ 3 0 نیا 6 -1 5
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. a b C d a-2b = 4c 2a = c + 3d
Find the coordinate vector [x]B of x relative to the given basis B = {b1,.....,bn}. b₁ = 4 3 ------- b₂ = 1 b3 = 7 5 X = 3
Find the coordinate vector [ x ]B of x relative to the given basis B = {b1,.....,bn}. 2 8 b₁ · = []=[] [] [] - 1 b₂ 4, b3 -2 X = 9 4 6
Determine which sets are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R3. Justify your answers. てー 0 [][]}] HH 0
Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers.All polynomials of the form p(t) = at2, where a is in R..
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. a b C d a + 3b = c b+c+a=d
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. {}] t S 5r1 s + 2t
Find the change-of-coordinates matrix from B to the standard basis in Rn. 2 = {-}] [] B=
Use an inverse matrix to find [ x ]B for the given x and B. 9- [ - ] = x + ( ²- ] [ - ] = 9 X
Find the change-of-coordinates matrix from B to the standard basis in Rn. L- (1H) --
Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers.All polynomials of the form p(t) = a + t2, where a is in R.
Use an inverse matrix to find [x]B for the given x and B. В = = {]- [9] ·x = [3] X=
Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers.All polynomials in Pn, such that p(0) = 0.
Assume that A is row equivalent to B. Find bases for Nul A, Col A, and Row A. -2 A = 2-6 T -3 468 11. -2 -4 -3 8 2-3 B = 1 0 6 5 0253 0000
Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers.All polynomials of degree at most 3, with integers as coefficients.
Leta. Is w in {V1, V2, V3}? How many vectors are in {V1, V2, V3}? b. How many vectors are in Span {V1, V2, V3}? c. Is w in the subspace spanned by {V1, V2, V3}? Why? --00-0 0, V2 = 1 , V3 2, and w=
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. c-6d d с : c, d real
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. b- 5d 26 2d + 1 d : b, d real
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. b-2d' 5 + d b + 3d d : b, d real
Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. -a + 2b a - 2b 3a-6b : a, b real
Assume that A is row equivalent to B. Find bases for Nul A, Col A, and Row A. A = B = 1 2 24 4 12 3 -5 11 -3 -5 -5 0 6 6-5 -5 2 05 15 4 1 00 -7 00 0 0 0 00 0 maing 04 2 19 -2 5 5 8 -9 0
Find A such that the given set is Col A. 2s + 3t r+s- 2t 4r + s 3r-s-t : r, s, t real
Find a basis for the set of vectors in R3 in the plane x + 4y - 5z = 0.
Find a basis for the space spanned by the given vectors, v1,....,v5. -3 2 -4 -3 OOOO 1 -8 6 7 1 2 -3 1 -6
Find a basis for the space spanned by the given vectors, v1,....,v5. 0 0 1 2 5 -3 3 -4 3
The set B = {1+t2, t + t2,1 + 2t + t2} is a basis for P2. Find the coordinate vector of p(t) = 1 + 4t + 7t2 relative to B.
Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is
Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is
Find A such that the given set is Col A. b-c 2b + c + d 5c - 4d d : b, c, d real
Find a basis for the space spanned by the given vectors, v1,....,v5. 8 9 -3 -6 0 5 1 -4 4 -4 -9 6 -7 6 8 4 -7 10 4 11 -8 -7
The set B = {1 - t2, t - t2, 2 - 2t + t2 is a basis for P2. Find the coordinate vector of p(t) = 3 + t - 6t2 relative to B.
Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is
Find a basis for the space spanned by the given vectors, v1,....,v5. 7 6 5 -7 8 -8 7 4 5 7] L- -7 -7 -9 -5 4 9 6 -91 3 -4 -1
For the matrices (a) Find k such that Nul A is a subspace of Rk, (b) Find k such that Col A is a subspace of Rk. A = = || 2-8 4 -4 −1 1
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) If x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rn.
Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is
Unless stated otherwise, B is a basis for a vector space V. Justify each answer.(T/F) If B is the standard basis for Rn, then the B-coordinate vector of an x in Rn is x itself.
The vectorsspan R2 but do not form a basis. Find two different ways to expressas a linear combination of v1, v2, v3. V1 • = [ - ] ] × = [ ³ ]- ~ = [ ³ ] V₂ V3 7
For the matrices (a) Find k such that Nul A is a subspace of Rk, (b) Find k such that Col A is a subspace of Rk. A = 4 5-2 1 1 0 6 1 0 07

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