A plane in R3 is represented parametrically by the equation * = w + su +...
Find a basis for the following plane in R3 1 + y - 2z = 0 First, solve the system, then assign parameters s and t to the free variables in this order), and write the solution in vector form as su + tv. Below, enter the components of the vectors u - (un, uz, uz)and v = (1, 0, vy)". ty and U-
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5. Parametrize the plane P in R3 containing the points x (1,0, ), x (2,0, 1) and x3 (1,3, 1). Does the plane P contain the point (-1,3,2)? 6. Sketch and parametrize the triangle in the plane with vertices x! = (-1,-2), x2 = (2, and x3 (1,3). Does this triangle contain the origin (0, 0,0)
4. Define an action of G = Symz on S = R3 as follows. For o € Symz and v= (21,22,23) ER3, set 0(0, 0) = (g(1), (2), g(3)). (a) Show that the subspace V = {(x1, x2, X3) | x1 + x2 + x3 = 0} has the following property: if v EV, then 0(0,v) is also in V. (b) Explain why the previous fact shows that Symz is isomorphic to a subgroup of GL2(R).
a 0 0 where a b, and c are positive numbers. Let S be the unit ball whose bounding surface has the equation x-x R3 + R3 be a linear transformation determined by the matrix A= 1 Complete Let 0 b 0 + x 0 0 c parts a and b below. u1 x1 2 ,2 2 a Show that T S is bounded by the ellipsoid with the equation 1 Create a vector u = that is within set...
Determine whether the system is consistent 1) x1 + x2 + x3 = 7 X1 - X2 + 2x3 = 7 5x1 + x2 + x3 = 11 A) No B) Yes Determine whether the matrix is in echelon form, reduced echelon form, or neither. [ 1 2 5 -7] 2) 0 1 -4 9 100 1 2 A) Reduced echelon form B) Echelon form C) Neither [1 0 -3 -51 300 1-3 4 0 0 0 0 LOO 0...
Problem #7: suppose that vectors in R3 are denoted by 1 x 3 matrices and define T: R3 R3 by 3 7([xi x2 x3]) = [x1 x2 x3]| 4 3 0 0] 8 Find a basis for the range of T. Problem #7: Select
Determine whether the set w is a subspace of R3 with the standard operations. If not, state why. (Select all that apply.) W = {(x1, 1/X1, X3): X1 and xy are real numbers, X1 + 0) W is a subspace of R W is not a subspace of R because it is not closed under addition. Wis not a subspace of R because it is not closed under scalar multiplication. X
Find an ONB (orthonormal basis) for the following plane in R3 2 + y + 3z = 0 First, solve the system, then assign parameters s and t to the free variables (in this order), and write the solution in vector form as su + tv. Now normalize u to have norm 1 and call it ū. Then find the component of v orthogonal to the line spanned by u and normalize it, call it ū. Below, enter the components...
0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1 = 19- and X 2 -- 2 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and normalize it...
Question 1 2 pts Describe the span of {(1,0,0),(0,0,1)} in R3 The x-z plane R3 R2 The x-y plane Question 2 2 pts Describe the span of {(1,1,1),(-1,-1, -1), (2,2, 2)} in R3 A plane passing through the origin Aline passing through the origin R3 A plane not passing through the origin A line not passing through the origin Question 3 2 pts Let u and v be vectors in R™ Then U-v=v.u True False Question 4 2 pts Ifu.v...