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(1 point) Do the following for the points (-3,1), (-2,2),(-1,1), (1,-2), (3,-2): (If you are entering...
0.5 -0.5 0.5 (1 point) Let A = -0.5 Note that the columns of A are...
0.5 -0.5 0.5 (1 point) Let A = -0.5 Note that the columns of A are orthonormal (why?). 0.5 0.5 0.5 0.5 -1 -2 (a) Solve the least squares problem Ax = b where b - -2 0 (b) Find the projection matrix P that projects vectors in R4 onto R(A) P = (c) Compute Ax and Pb Pb =
0.5 -0.5 0.5 (1 point) Let A = -0.5 Note that the columns of A are orthonormal (why?). 0.5 0.5...
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you...
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you wish to best-fit these to a line y = mx + b using least-squares. (a) Write down the corresponding matrix equation. (b) Solve for using the method of least squares. Make sure you simplify: the answer should not be complicated. (c) Find limin (d) The line corresponding to your answer in (c) passes through (3,2). Why does this make sense?
0.5 -0.5 0.5 -0.5 (1 point) Let A = . Note that the 0.5 0.5 0.5...
0.5 -0.5 0.5 -0.5 (1 point) Let A = . Note that the 0.5 0.5 0.5 0.5 columns of A are orthonormal (why?). (a) Solve the least squares problem Ax = b where b = Il (b) Find the projection matrix P that projects vectors in Ronto R(A) P= (c) Compute Ax and Pb Ax= Pb =
(1 point) Let A 0.5 -0.5 0.5 -0.5 0.5 0.5 0.5 0.5 Note that the columns...
(1 point) Let A 0.5 -0.5 0.5 -0.5 0.5 0.5 0.5 0.5 Note that the columns of A are orthonormal (why?). 3 2 (a) Solve the least squares problem Ax b where b 3 <X = (b) Find the projection matrix P that projects vectors in R* onto R(A) P (c) Compute Aî and Pb A Pb
2. This problem finds the curve C ++D = b which gives the best least squares...
2. This problem finds the curve C ++D = b which gives the best least squares fit to the points: t= -2, b=0 t = -1, b=0 t= 0, b=1 t= 1, b=1 t= 2, b=1 (a) (10 points) Write down the 5 equations Ax = b that would be satisfied if the curve went through all 5 points. (b) (10 points) Find the least squares solution = (Ĉ, Ð). (c) (10 points) Find the projection p of b onto...
(1 point) Are the following statements true or false? ? 1. The best approximation to y...
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...
== 2 1 3 (1 point) Let A 1 and b -3 2 6 The QR...
== 2 1 3 (1 point) Let A 1 and b -3 2 6 The QR factorization of the matrix A is given by: 1 2 = ſ v2 ŠV2 0 V2 3 2 3 (a) Applying the QR factorization to solving the least squares problem Ax b gives the system: 3 wls, wie X = (b) Use backsubstitution to system in part (a) and find the least squares solution. =
2. Let A:(-1,1,-1), B:(2,0.2), C:(4.1.-3), and D:(-3, 1, 10) be points in R. (a) Find the...
2. Let A:(-1,1,-1), B:(2,0.2), C:(4.1.-3), and D:(-3, 1, 10) be points in R. (a) Find the angles (in degrees) of the triangle with vertices A, B and C. (b) Find an equation of the plane passing through the points A, B, and C. (c) Find two unit vectors perpendicular the plane through A, B, and C. (d) Find the volume of the tetrahedron with vertices A, B, C, D. 3. (a) Find an equation of the tangent line to the...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QR and RS. (ii) Find the angle in degrees between QR and RS. (iii) Find ||QŘ|| and ||RŠI. (iv) Find the projection of R$ onto QR. 2. Let v = [6, 1, 2], w = [5,0,3), and P = (9,-7,31). (i) Find a vector u orthogonal to both v and w. (ii) Let L be...
1. find the derivative of f(×,y)=-4yx^3+xy^2 at P(1,1) in forward direction set by the line r(t)=(1+sqrt(2)t+sqrt(2)t...
1. find the derivative of f(×,y)=-4yx^3+xy^2 at P(1,1) in forward direction set by the line r(t)=(1+sqrt(2)t+sqrt(2)t 2. find an equation for the tangent plain at point P x^3+y^3=3xyz P(2,1,3/2)
0.5 -0.5 0.5 (1 point) Let A = -0.5 Note that the columns of A are orthonormal (why?). 0.5 0.5 0.5 0.5 -1 -2 (a) Solve the least squares problem Ax = b where b - -2 0 (b) Find the projection matrix P that projects vectors in R4 onto R(A) P = (c) Compute Ax and Pb Pb =
0.5 -0.5 0.5 (1 point) Let A = -0.5 Note that the columns of A are orthonormal (why?). 0.5 0.5...
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you wish to best-fit these to a line y = mx + b using least-squares. (a) Write down the corresponding matrix equation. (b) Solve for using the method of least squares. Make sure you simplify: the answer should not be complicated. (c) Find limin (d) The line corresponding to your answer in (c) passes through (3,2). Why does this make sense?
0.5 -0.5 0.5 -0.5 (1 point) Let A = . Note that the 0.5 0.5 0.5 0.5 columns of A are orthonormal (why?). (a) Solve the least squares problem Ax = b where b = Il (b) Find the projection matrix P that projects vectors in Ronto R(A) P= (c) Compute Ax and Pb Ax= Pb =
(1 point) Let A 0.5 -0.5 0.5 -0.5 0.5 0.5 0.5 0.5 Note that the columns of A are orthonormal (why?). 3 2 (a) Solve the least squares problem Ax b where b 3 <X = (b) Find the projection matrix P that projects vectors in R* onto R(A) P (c) Compute Aî and Pb A Pb
2. This problem finds the curve C ++D = b which gives the best least squares fit to the points: t= -2, b=0 t = -1, b=0 t= 0, b=1 t= 1, b=1 t= 2, b=1 (a) (10 points) Write down the 5 equations Ax = b that would be satisfied if the curve went through all 5 points. (b) (10 points) Find the least squares solution = (Ĉ, Ð). (c) (10 points) Find the projection p of b onto...
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...
== 2 1 3 (1 point) Let A 1 and b -3 2 6 The QR factorization of the matrix A is given by: 1 2 = ſ v2 ŠV2 0 V2 3 2 3 (a) Applying the QR factorization to solving the least squares problem Ax b gives the system: 3 wls, wie X = (b) Use backsubstitution to system in part (a) and find the least squares solution. =
2. Let A:(-1,1,-1), B:(2,0.2), C:(4.1.-3), and D:(-3, 1, 10) be points in R. (a) Find the angles (in degrees) of the triangle with vertices A, B and C. (b) Find an equation of the plane passing through the points A, B, and C. (c) Find two unit vectors perpendicular the plane through A, B, and C. (d) Find the volume of the tetrahedron with vertices A, B, C, D. 3. (a) Find an equation of the tangent line to the...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QR and RS. (ii) Find the angle in degrees between QR and RS. (iii) Find ||QŘ|| and ||RŠI. (iv) Find the projection of R$ onto QR. 2. Let v = [6, 1, 2], w = [5,0,3), and P = (9,-7,31). (i) Find a vector u orthogonal to both v and w. (ii) Let L be...
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