Let and consider V={x∈R^2 | Ax=5x}. Prove that V is a subspace of R^2, find a...
Let 
and consider
V={x∈R^2
| Ax=5x}. Prove that V
is a subspace of R^2, find a basis for
V, and determine its dimension.
Homework Answers
![* Given Itat A-[11] V = { XEvR) Ax =5x}. now, we have to show that visa subspace of 1R². so AX = 54, AY=57. A(x-y) = AX-AY =](http://img.homeworklib.com/questions/84062a60-3139-11eb-bce5-1b4f7f7f38a4.png?x-oss-process=image/resize,w_560)
![AX = 5x [97] [%] =5[3] [924) -5[*] = [0] (1928-59) - [O] » [ ***] = [] *1 *[]+4[*] = [] · Basis = {[7] [24]} dimension = num](http://img.homeworklib.com/questions/84968430-3139-11eb-b9af-3fa30e316a2e.png?x-oss-process=image/resize,w_560)
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Let and consider
V={x∈R^2
| Ax=5x}. Prove that V
is a subspace of R^2, find a...
Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 :...
Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 : 5x 2y z 0} a) Find a basis B for V. What is the dimension of V? b) Find a basis B' for R3 so that B C B'
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of...
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
For the rest of this problem, let V be a subspace of R" and let T:...
For the rest of this problem, let V be a subspace of R" and let T: R + R" be an orthogonal transformation such that T[V] = V1. (b) Prove that n is even and that dim V = dimV+ = (c) Prove that T[v+] = V. (d) Prove that there is a basis B of R" such that the B-matrix of T has block form (T) = [% ] where Qi and Q2 are orthogonal matrices,
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of...
Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of R". (b) Prove that if a vector v belongs to both W and W+, then v must be the zero vector.
How can I get the (a) 3*2 matrix A? x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0...
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace...
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B=
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0},...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX)...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
(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}...
Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 : 5x 2y z 0} a) Find a basis B for V. What is the dimension of V? b) Find a basis B' for R3 so that B C B'
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
For the rest of this problem, let V be a subspace of R" and let T: R + R" be an orthogonal transformation such that T[V] = V1. (b) Prove that n is even and that dim V = dimV+ = (c) Prove that T[v+] = V. (d) Prove that there is a basis B of R" such that the B-matrix of T has block form (T) = [% ] where Qi and Q2 are orthogonal matrices,
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of R". (b) Prove that if a vector v belongs to both W and W+, then v must be the zero vector.
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B=
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
(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}...
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