Let L in R 3 be the line through the origin
spanned by the vector v = 1 1 3 . Find the linear equations
that define L, i.e., find a system of linear equations whose
solutions are the points in L.
Homework Answers
The equation of L is,
(x-0)/1 = (y-0)/1 = (z-0)/3
i.e. x/1 = y/1 = z/3
Or, x = y = z/3
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Let L in R 3 be the line through the origin
spanned by the vector v...
Let L in R 3 be the line through the origin spanned by the vector v...
Let L in R 3 be the line through the origin spanned by the
vector v = 1 1 3 . Find the linear equations that define L,
i.e., find a system of linear equations whose solutions are the
points in L. (7) Give an example of a linear transformation from T
: R 2 → R 3 with the following two properties: (a) T is not
one-to-one, and (b) range(T) = ...
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...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
6. Let L be the line in spanned by the vector u =(1,-1,2). (a) (6 points)...
6. Let L be the line in spanned by the vector u =(1,-1,2). (a) (6 points) Compute a basis for the subspace Zt. 7. (6 point bonus! Find the general solution y to the second-order linear differential equa- tion below. Use C.C.C.... for the names of any unknown constants. 0-1 + 424 = 0 (b) (6 points Use the Gram-Schuit process to find an orthonormal basis for L,
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t...
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t +1,4, 2t) be a parametrized curve. Compute the point r(to) on the curve which is closest to l, and state the distance from r(to) to l.
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.
(6) In R3, let W be the set of solutions of the homogeneous linear equation r...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx...
3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 7z = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working. 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....
1. Let Q = (-3, -3, -3.3), R = (-3, -3, -33) and S = (1,...
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 QŘ and RS. (ii) Find ||QR|| and ||RŠI. (iii) Find the angle in degrees between QR and RS. (iv) Find the projection of RŠ onto QŘ. 2. Let v= [6, 1, 2], w = [5,0,3], and P = (9, -7,31). (i) Find a vector u orthogonal to both v...
2. Let v= [6, 1, 2], w = [5,0, 3), and P= (9, -7,31). (i) Find...
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 the line in R3 that passes through the point P and is perpendicular to both of the vectors v and w. Find an equation for the line L in vector form. (iii) Find parametric equations for the line L.
Let L in R 3 be the line through the origin spanned by the
vector v = 1 1 3 . Find the linear equations that define L,
i.e., find a system of linear equations whose solutions are the
points in L. (7) Give an example of a linear transformation from T
: R 2 → R 3 with the following two properties: (a) T is not
one-to-one, and (b) range(T) = ...
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...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
6. Let L be the line in spanned by the vector u =(1,-1,2). (a) (6 points) Compute a basis for the subspace Zt. 7. (6 point bonus! Find the general solution y to the second-order linear differential equa- tion below. Use C.C.C.... for the names of any unknown constants. 0-1 + 424 = 0 (b) (6 points Use the Gram-Schuit process to find an orthonormal basis for L,
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t +1,4, 2t) be a parametrized curve. Compute the point r(to) on the curve which is closest to l, and state the distance from r(to) to l.
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 7z = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working. 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....
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 QŘ and RS. (ii) Find ||QR|| and ||RŠI. (iii) Find the angle in degrees between QR and RS. (iv) Find the projection of RŠ onto QŘ. 2. Let v= [6, 1, 2], w = [5,0,3], and P = (9, -7,31). (i) Find a vector u orthogonal to both v...
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 the line in R3 that passes through the point P and is perpendicular to both of the vectors v and w. Find an equation for the line L in vector form. (iii) Find parametric equations for the line L.
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