Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection...
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x
(a) Compute R(x) for x=(1260)
(b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
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Let E be the plane in R3 spanned by the orthogonal vectors
v1=(121)and v2=(−11−1) The reflection...
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1...
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
Let S be the tetrahedron in R3 with vertices at x the vectors 0, e1, e2, and e3, and let S' be the tetrahedron with...
Let S be the tetrahedron in R3 with vertices at x the vectors 0, e1, e2, and e3, and let S' be the tetrahedron with vertices at vectors 0, v1, V2 and v3. See the figures to the right. Complete parts (a) and (b) below. a. Describe a linear transformation that maps S onto S lf T is a linear transformation that maps S onto S, then the standard matrix for T, written in terms of v1-v2, and v3, is...
Problem 13. Let l be the line in R' spanned by the vector u = 3...
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
Problem 6. Let E be the plane: 2xi- x2 x3 = 0, and let P R3R3 be the orthogonal _ projection onto the plane E. Let v 1...
Problem 6. Let E be the plane: 2xi- x2 x3 = 0, and let P R3R3 be the orthogonal _ projection onto the plane E. Let v 1 (1) What are the image and kernel of P? What is the rank of P? Give a geometric descrip- tion, without relying (2) Give four different vectors e R3 such that Px Pv. (Again, solve geometrically and do not use the matrix of P.) (3) Find Pv (4) Find the reflection of...
חו (1 point) Suppose V1, V2, V3 is an orthogonal set of vectors in R Let...
חו (1 point) Suppose V1, V2, V3 is an orthogonal set of vectors in R Let w be a vector in span(V1, V2, V3) such that (v1,vi) = 24, (v2,v2) = 21, (V3, V3) = 9, (w,v) 120, (w, v2) = 147, (w,v3) -36, Vi+ V2+ then w= V3.
Please answer the following questions with clear working out. 09. 3. (a) Let M- (i) Find...
Please answer the following questions with clear working out.
09.
3. (a) Let M- (i) Find the eigenspace of M corresponding to the eigenvalue -1. (ii) A linear transformation T : R2 R2 is defined by T ((3 )) M ( 5) for all ?ER2 Which straight lines through the origin in R2 are fixed by T? 2 Let Vi = (-1 and V2- (i) Explain why {vi, V2 is a basis for R2 (ii) Write (i) as a linear...
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11...
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11 a) Find the eigenvalues of A b) For each eigenvalue of A, find a basis for the corresponding eigenspace. c) Consider the linear transformation T : R3 -> R3 defined by T(x) = Ax for every xE R3. Find a basis of R3 in which the matrix representing T is diagonal
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R",...
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R", and let T:R" → R" be a linear transformation such that the set {T(01), T(V2), ...,T(Un) } is basis for R". Show that B = {01, V2, ..., Un } is also a basis for R". Problem 7: Decide whether the following statement is true or false. If it is true, prove it. If it is false, give an example to show that it...
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers...
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers dı, d2, d3 € R. (i) Show that there is a matrix A E M3(R), and only one, with eigenvalues dı, d2, d3 and corresponding eigenvectors V1, V2, V3. (ii) Show that if {V1, V2, V3} is an orthonormal set of vectors. then the matrix A is symmetric.
(1 point) Suppose V1, V2, U3 is an orthogonal set of vectors in R. Let w...
(1 point) Suppose V1, V2, U3 is an orthogonal set of vectors in R. Let w be a vector in Span(V1, U2, U3) such that 01.01 = 33, U2 · U2 = 10.25, 03 · 03 = 36, W • V1 = 99, w · U2 = 71.75, w · Uz = -108, then w = Vi+ U2+ U3
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
Let S be the tetrahedron in R3 with vertices at x the vectors 0, e1, e2, and e3, and let S' be the tetrahedron with vertices at vectors 0, v1, V2 and v3. See the figures to the right. Complete parts (a) and (b) below. a. Describe a linear transformation that maps S onto S lf T is a linear transformation that maps S onto S, then the standard matrix for T, written in terms of v1-v2, and v3, is...
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
Problem 6. Let E be the plane: 2xi- x2 x3 = 0, and let P R3R3 be the orthogonal _ projection onto the plane E. Let v 1 (1) What are the image and kernel of P? What is the rank of P? Give a geometric descrip- tion, without relying (2) Give four different vectors e R3 such that Px Pv. (Again, solve geometrically and do not use the matrix of P.) (3) Find Pv (4) Find the reflection of...
חו (1 point) Suppose V1, V2, V3 is an orthogonal set of vectors in R Let w be a vector in span(V1, V2, V3) such that (v1,vi) = 24, (v2,v2) = 21, (V3, V3) = 9, (w,v) 120, (w, v2) = 147, (w,v3) -36, Vi+ V2+ then w= V3.
Please answer the following questions with clear working out.
09.
3. (a) Let M- (i) Find the eigenspace of M corresponding to the eigenvalue -1. (ii) A linear transformation T : R2 R2 is defined by T ((3 )) M ( 5) for all ?ER2 Which straight lines through the origin in R2 are fixed by T? 2 Let Vi = (-1 and V2- (i) Explain why {vi, V2 is a basis for R2 (ii) Write (i) as a linear...
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11 a) Find the eigenvalues of A b) For each eigenvalue of A, find a basis for the corresponding eigenspace. c) Consider the linear transformation T : R3 -> R3 defined by T(x) = Ax for every xE R3. Find a basis of R3 in which the matrix representing T is diagonal
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R", and let T:R" → R" be a linear transformation such that the set {T(01), T(V2), ...,T(Un) } is basis for R". Show that B = {01, V2, ..., Un } is also a basis for R". Problem 7: Decide whether the following statement is true or false. If it is true, prove it. If it is false, give an example to show that it...
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers dı, d2, d3 € R. (i) Show that there is a matrix A E M3(R), and only one, with eigenvalues dı, d2, d3 and corresponding eigenvectors V1, V2, V3. (ii) Show that if {V1, V2, V3} is an orthonormal set of vectors. then the matrix A is symmetric.
(1 point) Suppose V1, V2, U3 is an orthogonal set of vectors in R. Let w be a vector in Span(V1, U2, U3) such that 01.01 = 33, U2 · U2 = 10.25, 03 · 03 = 36, W • V1 = 99, w · U2 = 71.75, w · Uz = -108, then w = Vi+ U2+ U3
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