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(1 point) Consider the line L : X2 =-4x1- a. Find the matrix A which projects...
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below:...
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction. (c) L doubles the length of t and then rotates it 30° in the counterclockwise direction. (d) L reflects each vector 7 about the line x2 = 21 and projects it onto the...
(1 point) Find the matrix A of the orthogonal projection onto the line L in R2...
(1 point) Find the matrix A of the orthogonal projection onto the line L in R2 that consists of all scalar multiples of the vector [63].
22. (a) Find two vectors that span the null space of A 3 -1 2 -4 (b) Use the result of part (a) to find the matrix that projects vectors onto the null space of A. (c) Find two orthogonal vectors...
22. (a) Find two vectors that span the null space of A 3 -1 2 -4 (b) Use the result of part (a) to find the matrix that projects vectors onto the null space of A. (c) Find two orthogonal vectors that span the null space of A. (d) Use the result of (c) to find the matrix that projects vectors onto the nul space of A. Compare this matrix with the one found in part (a). (e) Find the...
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric...
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
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...
Consider the linear system x1 +4x2 = 0 4x1 +x2 = 0 The true solution is...
Consider the linear system x1 +4x2 = 0 4x1 +x2 = 0 The true
solution is x1 = ?1=15, x2 = 4=15. Apply the Jacobi and
Gauss-Seidel methods with x(0) = [0; 0]T to the system and nd out
which methods diverge more rapidly. Next, interchange the two
equations to write the system as 8< : 4x1 +x2 = 0 x1 +4x2 = 0
and apply both methods with x(0) = [0; 0]T . Iterate until
jjx?x(k)jj 10?5. Which method...
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which...
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ?
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of projection) is located at the point (0, 0, 11). Using homogeneous coordinates, a perspective projection matri A which projects the onto the xy-plane is given by Consider a triangle with comers at A (5,4,4), B (1,6,3), and C (3,3,7). Projecting this onto the xy-plane under this transformation gives a triangle with corners at the points A', B', and C' correspondingly where C' (Express...
(1 point) Let L be the linear operator in R? defined by L(x) = (4x1 –...
(1 point) Let L be the linear operator in R? defined by L(x) = (4x1 – 2x2, -6x1 + 3x2) Find bases of the kernel and image of L. 00 Kernel: * Image: [-2,3] To enter a basis into WebWork, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is 1 2,1l/, then you would enter [1,2,3], 31 [1,1,1) into the answer blank.
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2....
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2. Hint: What is the matrix of the projection onto the coordinate axis x1? Problem 5. Problem 5.4 on page 23. The following method is suggested: (1) Find an angle o such that the line x1 = –2x2 is obtained by rotating the x-axis by 0. (2) Convince yourself with geometry that to project a vector v onto the line x1 = –2x2 is the...
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction. (c) L doubles the length of t and then rotates it 30° in the counterclockwise direction. (d) L reflects each vector 7 about the line x2 = 21 and projects it onto the...
22. (a) Find two vectors that span the null space of A 3 -1 2 -4 (b) Use the result of part (a) to find the matrix that projects vectors onto the null space of A. (c) Find two orthogonal vectors that span the null space of A. (d) Use the result of (c) to find the matrix that projects vectors onto the nul space of A. Compare this matrix with the one found in part (a). (e) Find the...
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
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...
Consider the linear system x1 +4x2 = 0 4x1 +x2 = 0 The true
solution is x1 = ?1=15, x2 = 4=15. Apply the Jacobi and
Gauss-Seidel methods with x(0) = [0; 0]T to the system and nd out
which methods diverge more rapidly. Next, interchange the two
equations to write the system as 8< : 4x1 +x2 = 0 x1 +4x2 = 0
and apply both methods with x(0) = [0; 0]T . Iterate until
jjx?x(k)jj 10?5. Which method...
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ?
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of projection) is located at the point (0, 0, 11). Using homogeneous coordinates, a perspective projection matri A which projects the onto the xy-plane is given by Consider a triangle with comers at A (5,4,4), B (1,6,3), and C (3,3,7). Projecting this onto the xy-plane under this transformation gives a triangle with corners at the points A', B', and C' correspondingly where C' (Express...
(1 point) Let L be the linear operator in R? defined by L(x) = (4x1 – 2x2, -6x1 + 3x2) Find bases of the kernel and image of L. 00 Kernel: * Image: [-2,3] To enter a basis into WebWork, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is 1 2,1l/, then you would enter [1,2,3], 31 [1,1,1) into the answer blank.
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2. Hint: What is the matrix of the projection onto the coordinate axis x1? Problem 5. Problem 5.4 on page 23. The following method is suggested: (1) Find an angle o such that the line x1 = –2x2 is obtained by rotating the x-axis by 0. (2) Convince yourself with geometry that to project a vector v onto the line x1 = –2x2 is the...
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