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(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u,...
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e...
hint:
H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
(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...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear of the u's Which of the following cniteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of IR? Select all that apply A. The vectors must span W B. The vectors must all have a length of 1 D C. The distance between any pair of distinct vectors must be constant D. The vectors must form...
(11 Let u Show that B } is an orthogonal basis of R3. (b) Convert B...
(11 Let u Show that B } is an orthogonal basis of R3. (b) Convert B into an orthonormal basis C of R3 by normalizing ü, ū and w. Show your work. Find the change of coordinates matrices Psee and Pee-swhere C is the or- thonormal basis of R3 you found in (b) and S is the standard basis of R3. Justify your answers. Suppose now that ü, ū and w are eigenvectors of a 3 x 3 matrix A...
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 sta...
Hi,
could you post solutions to the following questions. Thanks.
2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and...
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and that the vector Ū E span{ū1, ... , ūk}. Then û = Proj, Ū = ū. True O False Suppose that W is a subspace of R" and that the vector ŪER" .Then if û = Projű we have Ilu - Oll < 110 - ūll for all vectors ū EW . That is, <- is the vector in W that is closest to...
Please do only e and f and show work null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Fin...
Please do only e and f and show work
null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Find bases for the four fundamental subspaces of 1 1 1 6 -1 0 1 -1 2 A= -2 3 1 -2 1 4 1 6 1 3 8. Given a subspace W of R", define the orthogonal complement of W to be W vE R u v 0 for every u E W (a) Let W span(e, e2)...
Suppose that {ūj, ..., ūk} is an orthonormal basis for a subspace W of R" and...
Suppose that {ūj, ..., ūk} is an orthonormal basis for a subspace W of R" and we form the matrix U = (ū; ū2 ... ük) Then the matrix P= UUT has the property that p2 = P . This follows for the following reason(s). A. We know that P= I and so P2 = 1? = I = P B. We can calculate p2 = (UUT) (UUT) = U (UTU) UT = UIUT C. Since P is a projection...
Let V = M2(R), and let U be the span of S = 2. (a) Let...
Let V = M2(R), and let U be the span of
S =
2. (a) Let V = M,(R), and let U be the span of s={(1 1) ($ 3). (3), (1 9). (1) 2.)} Find a basis for U contained in S. (b) Let W be the subspace of P spanned by T = {2} + 22 – 1, -2.3 + 2x +1,23 +22² + 2x – 1, 2x3 + x2 +1 -2, 4.23 + 2x2 - -4}. Find...
I am looking for how to explain #4 part b. I have gotten the matrix A...
I am looking for how to explain #4 part b. I have gotten the
matrix A and I believe the answer is W = span{ v1 u2 u3 } however
I'm not really sure if that is correct or not. Please give a small
explanation. Also im not sure if I need to represent the vectors in
A as columns or rows, or if either one works.
For the next two problems, W is the subspace of R4 given by...
hint:
H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
(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...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear of the u's Which of the following cniteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of IR? Select all that apply A. The vectors must span W B. The vectors must all have a length of 1 D C. The distance between any pair of distinct vectors must be constant D. The vectors must form...
(11 Let u Show that B } is an orthogonal basis of R3. (b) Convert B into an orthonormal basis C of R3 by normalizing ü, ū and w. Show your work. Find the change of coordinates matrices Psee and Pee-swhere C is the or- thonormal basis of R3 you found in (b) and S is the standard basis of R3. Justify your answers. Suppose now that ü, ū and w are eigenvectors of a 3 x 3 matrix A...
Hi,
could you post solutions to the following questions. Thanks.
2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and that the vector Ū E span{ū1, ... , ūk}. Then û = Proj, Ū = ū. True O False Suppose that W is a subspace of R" and that the vector ŪER" .Then if û = Projű we have Ilu - Oll < 110 - ūll for all vectors ū EW . That is, <- is the vector in W that is closest to...
Please do only e and f and show work
null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Find bases for the four fundamental subspaces of 1 1 1 6 -1 0 1 -1 2 A= -2 3 1 -2 1 4 1 6 1 3 8. Given a subspace W of R", define the orthogonal complement of W to be W vE R u v 0 for every u E W (a) Let W span(e, e2)...
Suppose that {ūj, ..., ūk} is an orthonormal basis for a subspace W of R" and we form the matrix U = (ū; ū2 ... ük) Then the matrix P= UUT has the property that p2 = P . This follows for the following reason(s). A. We know that P= I and so P2 = 1? = I = P B. We can calculate p2 = (UUT) (UUT) = U (UTU) UT = UIUT C. Since P is a projection...
Let V = M2(R), and let U be the span of
S =
2. (a) Let V = M,(R), and let U be the span of s={(1 1) ($ 3). (3), (1 9). (1) 2.)} Find a basis for U contained in S. (b) Let W be the subspace of P spanned by T = {2} + 22 – 1, -2.3 + 2x +1,23 +22² + 2x – 1, 2x3 + x2 +1 -2, 4.23 + 2x2 - -4}. Find...
I am looking for how to explain #4 part b. I have gotten the
matrix A and I believe the answer is W = span{ v1 u2 u3 } however
I'm not really sure if that is correct or not. Please give a small
explanation. Also im not sure if I need to represent the vectors in
A as columns or rows, or if either one works.
For the next two problems, W is the subspace of R4 given by...
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