Homework Answers
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Problem 2 [10 pts] For a fixed matrices B, C e R2x2, let W = {1...
Problem 2 [10 pts] For a fixed matrices B, C e R2x2, let W = {1 € R2x2 | A*B=24*C}. Determine if W is a subspace of R2x2 (either prove that it is, or show via specific counterexample that it isn't).
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,):...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
Let W be the set of singular (noninvertible) matrices of order 2. Show that W is...
Let W be the set of singular (noninvertible) matrices
of order 2. Show that W is not a subspace of
M2×2 with the standard matrix operations.
Q1: Let W be the set of singular (noninvertible) matrices of order 2. Show that W is not a subspace of M2x2 with the standard matrix operations.-
Problem 3. Let V and W be vector spaces, let T : V -> W be...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
-9 2. Let Vi-8.V2,andvs-2, let B -(V,V2,Vs), and let W be the subspace spanned , let B -(Vi,V2,V3...
-9 2. Let Vi-8.V2,andvs-2, let B -(V,V2,Vs), and let W be the subspace spanned , let B -(Vi,V2,V3), and let W be the subspace spanned by B. Note that B is an orthogonal set. 17 a. 1 point] Find the coordinates of uwith respect to B, without inverting any matrices or L-2 solving any systems of linear equations. 35 16 25 b. 1 point Find the orthogonal projection of to W, without inverting any matrices or solving any systems of...
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries....
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries. Let W M2x2 (R) det (A) 0. Show that W is not a subspace of M2x2(R) A E
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3}...
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3} C R6. 11 9 1 2 a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W. You should explicitly show each step of your calculation. 10 -7 11 b) [5 pts) Let v = Compute the projection prw(v) of v onto the subspace W using the 5 orthogonal basis in a). c) (4 pts] Use the computation in b) to...
SVD a) Let A E RX be an invertible matrix and i ER" be a nonzero...
SVD a) Let A E RX be an invertible matrix and i ER" be a nonzero vector. Prove that ||A7|| 2 min ||- b) Let A € R2X2 and 1 = plot of|ly|| vse. 2,17|| = 1. Now let y = Až. Below is the (cos(O)" A has the SVDUEVT. Either specify what the matrices U, 2, and V are; or state they they cannot be determined from the information given. c) Let A E RNXN,B E RNXN be full...
Problem #9: [2 marks] Let W be the the subspace of M14.14 (i.e., 14 x 14...
Problem #9: [2 marks] Let W be the the subspace of M14.14 (i.e., 14 x 14 matrices) consisting of all lower triangular matrices. Find the dimension of W. Problem #9:
Help! Let B and C be similar nxn matrices. Prove that the matrices given by: I...
Help!
Let B and C be similar nxn matrices. Prove that the matrices given by: I +5B - 2B4 and I +5C - 204 are similar. (6 pts)
Problem 2 [10 pts] For a fixed matrices B, C e R2x2, let W = {1 € R2x2 | A*B=24*C}. Determine if W is a subspace of R2x2 (either prove that it is, or show via specific counterexample that it isn't).
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
Let W be the set of singular (noninvertible) matrices
of order 2. Show that W is not a subspace of
M2×2 with the standard matrix operations.
Q1: Let W be the set of singular (noninvertible) matrices of order 2. Show that W is not a subspace of M2x2 with the standard matrix operations.-
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
-9 2. Let Vi-8.V2,andvs-2, let B -(V,V2,Vs), and let W be the subspace spanned , let B -(Vi,V2,V3), and let W be the subspace spanned by B. Note that B is an orthogonal set. 17 a. 1 point] Find the coordinates of uwith respect to B, without inverting any matrices or L-2 solving any systems of linear equations. 35 16 25 b. 1 point Find the orthogonal projection of to W, without inverting any matrices or solving any systems of...
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries. Let W M2x2 (R) det (A) 0. Show that W is not a subspace of M2x2(R) A E
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3} C R6. 11 9 1 2 a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W. You should explicitly show each step of your calculation. 10 -7 11 b) [5 pts) Let v = Compute the projection prw(v) of v onto the subspace W using the 5 orthogonal basis in a). c) (4 pts] Use the computation in b) to...
SVD a) Let A E RX be an invertible matrix and i ER" be a nonzero vector. Prove that ||A7|| 2 min ||- b) Let A € R2X2 and 1 = plot of|ly|| vse. 2,17|| = 1. Now let y = Až. Below is the (cos(O)" A has the SVDUEVT. Either specify what the matrices U, 2, and V are; or state they they cannot be determined from the information given. c) Let A E RNXN,B E RNXN be full...
Problem #9: [2 marks] Let W be the the subspace of M14.14 (i.e., 14 x 14 matrices) consisting of all lower triangular matrices. Find the dimension of W. Problem #9:
Help!
Let B and C be similar nxn matrices. Prove that the matrices given by: I +5B - 2B4 and I +5C - 204 are similar. (6 pts)
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