Homework Answers
2. (5' each) Give short answers: (a) True or false: If Ai-Adi for some real number λ, then u is a...
linear algebra question
2. (5' each) Give short answers: (a) True or false: If Ai-Adi for some real number λ, then u is an eigenvector of matrix A. If a square matrix is diagonalizable, then it has n distinct real eigenvalues. Two vectors of the same dimension are linearly independent if and only if one is not a multiple of the other. If the span of a set of vectors is R", then that set is a basis of R...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
3. Let B ERnxn be a symmetrie and P.D. matrix. Show that l s (o Bu) (B-v) for any nonzero v E R",...
3. Let B ERnxn be a symmetrie and P.D. matrix. Show that l s (o Bu) (B-v) for any nonzero v E R", and that the equality holds if and only if v is an eigenvector of B. (Hnt: note that llt -W/2t, B-1/2v), and use the Cauchy-Schwarz inequality.) 4. Let (ak) be a real sequence such that for each k, either akil > ak or akt? where, is a constant independent of k. Show that a 2 min(ai, T)...
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W! Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] L...
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
1. Find a matrix A so that A | y for all z, y, z E...
1. Find a matrix A so that A | y for all z, y, z E R. What are the dimensions of A? 2y +2z (The dimensions of an m x n matrix are "m × n.) for all R2. Find a matrix A so that T-LA (that is. Τ(x) = Ax for all fe R2). and all vectorsR2. Do not assume any properties of the dot product, beyond the definition. (Hint write Aa21 a22and x 2. Let T: IR2R2...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
1 0 0 0 0 I. (2 points each item) Let R=13 0 1 0 0...
1 0 0 0 0 I. (2 points each item) Let R=13 0 1 0 0 (a) Let A be any other matrix with 5 rows. Explain in words what multiplying A on the left by R does to the rows of A (b) Explain in words what you would do to undo the effect of multiplying A on the left by R (c) Without doing any computations, write down the inverse of R (d) Note that (RTA)T-AT((RT)T-ATR. Let C...
2. Given the columns of the matrix u v w 0 1 2 0-1 0 0 r S t -1 021 01 0 For each of the sets of ...
Please solve using matrices and not equations.
Thanks.
2. Given the columns of the matrix u v w 0 1 2 0-1 0 0 r S t -1 021 01 0 For each of the sets of vectors given below, answer the following questions: (i) Is the set linearly independent? 1 Does the set span (iii Does the vector a- (a) S (r, s, t, u) (b) T fr,t, 0, u) (c) U = {r, t, w, u, v} (3,2,1,5)...
Question 1 Question 2 Let u, v, w be three vectors in R4 with the property...
Question 1
Question 2
Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
my solutions say linearly independent but i dont understand why 4. (5 pts) Let zu(e) =...
my solutions say linearly independent but i dont understand
why
4. (5 pts) Let zu(e) = (2-1), sz(t) = [et] Determine whether the vector functions are linearly dependent or linearly independent on (-0,00). ww/xix.7(4) = fet to +-+-0 W[X, Xz] (t) = 0
linear algebra question
2. (5' each) Give short answers: (a) True or false: If Ai-Adi for some real number λ, then u is an eigenvector of matrix A. If a square matrix is diagonalizable, then it has n distinct real eigenvalues. Two vectors of the same dimension are linearly independent if and only if one is not a multiple of the other. If the span of a set of vectors is R", then that set is a basis of R...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
3. Let B ERnxn be a symmetrie and P.D. matrix. Show that l s (o Bu) (B-v) for any nonzero v E R", and that the equality holds if and only if v is an eigenvector of B. (Hnt: note that llt -W/2t, B-1/2v), and use the Cauchy-Schwarz inequality.) 4. Let (ak) be a real sequence such that for each k, either akil > ak or akt? where, is a constant independent of k. Show that a 2 min(ai, T)...
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
1. Find a matrix A so that A | y for all z, y, z E R. What are the dimensions of A? 2y +2z (The dimensions of an m x n matrix are "m × n.) for all R2. Find a matrix A so that T-LA (that is. Τ(x) = Ax for all fe R2). and all vectorsR2. Do not assume any properties of the dot product, beyond the definition. (Hint write Aa21 a22and x 2. Let T: IR2R2...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
1 0 0 0 0 I. (2 points each item) Let R=13 0 1 0 0 (a) Let A be any other matrix with 5 rows. Explain in words what multiplying A on the left by R does to the rows of A (b) Explain in words what you would do to undo the effect of multiplying A on the left by R (c) Without doing any computations, write down the inverse of R (d) Note that (RTA)T-AT((RT)T-ATR. Let C...
Please solve using matrices and not equations.
Thanks.
2. Given the columns of the matrix u v w 0 1 2 0-1 0 0 r S t -1 021 01 0 For each of the sets of vectors given below, answer the following questions: (i) Is the set linearly independent? 1 Does the set span (iii Does the vector a- (a) S (r, s, t, u) (b) T fr,t, 0, u) (c) U = {r, t, w, u, v} (3,2,1,5)...
Question 1
Question 2
Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
my solutions say linearly independent but i dont understand
why
4. (5 pts) Let zu(e) = (2-1), sz(t) = [et] Determine whether the vector functions are linearly dependent or linearly independent on (-0,00). ww/xix.7(4) = fet to +-+-0 W[X, Xz] (t) = 0
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