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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...
In Problems 1-9, consider the given vector x. Find the vectors that result from each of the follo...
1,5
In Problems 1-9, consider the given vector x. Find the vectors that result from each of the following: (a) stretch by a factor of c (sketch the original vector and the resulting vector) (b) rotation by an angle of ф (sketch the original vector, the angle of rotation, 716 Appendix B. Selected Topics from Linear Algebra and the resulting vector) original vector, the line of projection, and the resulting vector) the original vector, the line of reflection, and the...
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The proj...
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The projection of R3 onto L — denoted PL : R3 −→ R3 — is definded to be equal to the projection pu onto the vector u. You may assume that PL is a linear transformation. Find the standard matrix [PL] for PL.
(1) (Definition and short answer — no justification needed) (a) Let f:R → R", and let...
(1) (Definition and short answer — no justification needed) (a) Let f:R → R", and let p ER". Define carefully what it means for the function f to be differentiable at p. (b) Given a linear transformation T : R" + R", explain briefly how to form its representing matrix (T). If you know the matrix (T), how can you compute T(v) for a vector v € R"? 1 and let S be the linear (c) Let T be the...
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...
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...
Let A be the 3 x 3 matrix such that, for any. ✓ E R', Av...
Let A be the 3 x 3 matrix such that, for any. ✓ E R', Av gives the projection of ū onto the plane x + y + z = 0. Determine A15
3. Find m for which the following lines do not form a triangle. x+2y 5D, 2x-3y-4 .2, mx+y 0 [Sol] Since line 3 pass...
3. Find m for which the following lines do not form a triangle. x+2y 5D, 2x-3y-4 .2, mx+y 0 [Sol] Since line 3 passes through the origin, lines D, 2 and 3will not form a triangle in the following three cases: wor When D and 3 are (i) parallel. doa When 2 and 3 are (ii) parallel. When D, and 3 all intersect at one point. (ii Therefore, from ( i ), (ii) and (iii), m 4. Find a for...
Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n ...
Please show all work in READ-ABLE way. Thank you so much in
advance.
Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector...
(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector 90 degrees counterclockwise. Let A be the standard matrix for T. Is A diagonalizable over R? What about over C? (e) (3 points) Let T : R4 → R4 be given by T(x) = Ax, A = 3 -1 7 12 0 0 0 4 0 0 5 4 0 4 2 1 Is E Im(T)? 3 (f) (9 points) Let U be a...
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...
1,5
In Problems 1-9, consider the given vector x. Find the vectors that result from each of the following: (a) stretch by a factor of c (sketch the original vector and the resulting vector) (b) rotation by an angle of ф (sketch the original vector, the angle of rotation, 716 Appendix B. Selected Topics from Linear Algebra and the resulting vector) original vector, the line of projection, and the resulting vector) the original vector, the line of reflection, and the...
(1) (Definition and short answer — no justification needed) (a) Let f:R → R", and let p ER". Define carefully what it means for the function f to be differentiable at p. (b) Given a linear transformation T : R" + R", explain briefly how to form its representing matrix (T). If you know the matrix (T), how can you compute T(v) for a vector v € R"? 1 and let S be the linear (c) Let T be the...
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...
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...
Let A be the 3 x 3 matrix such that, for any. ✓ E R', Av gives the projection of ū onto the plane x + y + z = 0. Determine A15
3. Find m for which the following lines do not form a triangle. x+2y 5D, 2x-3y-4 .2, mx+y 0 [Sol] Since line 3 passes through the origin, lines D, 2 and 3will not form a triangle in the following three cases: wor When D and 3 are (i) parallel. doa When 2 and 3 are (ii) parallel. When D, and 3 all intersect at one point. (ii Therefore, from ( i ), (ii) and (iii), m 4. Find a for...
Please show all work in READ-ABLE way. Thank you so much in
advance.
Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector 90 degrees counterclockwise. Let A be the standard matrix for T. Is A diagonalizable over R? What about over C? (e) (3 points) Let T : R4 → R4 be given by T(x) = Ax, A = 3 -1 7 12 0 0 0 4 0 0 5 4 0 4 2 1 Is E Im(T)? 3 (f) (9 points) Let U be a...
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