Find vectors V1, ..., VKE R3 that span the plane in R3 with equation x-2y+32 =...
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
(4) Find the span of the vectors You answer should be either: 0}, a 3 1 2 line through the origin, a plane through the origin, or R3. Determine which one it is. If it's a line or a plane, find its equation
Question 1 (10 points) Projection matrix and Normal equation: Consider the vectors v1 = (1, 2, 1), V2 = (2,4, 2), V3 = (0,1,0), and v4 = (3, 7,3). (a) (2 points) Obtain a basis for R3 that includes as many of these vectors as possible. (b) (4 points) Obtain the orthogonal projection matrices onto the plane V = span{v1, v3} and its perpendicular complement V+. (c) (2 points) Use this result to decompose the vector b= (-1,1,1) into a...
Problem 1. Let V1,...,V5 be vectors in R3, as shown in the four figures below. In each figure, find all linearly dependent sets consisting of three of these five vectors, or else state that there are none if this is the case. No justification needed. (Note that in each of these figures, Vi and V2 span the displayed plane, V3 points "up" and is perpendicular to this plane, and for any other vector not in the plane, we draw a...
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...
Let H = Span{V1, V2} and K = Span{V3,V4}, where V1, V2, V3, and V4 are given below. 1 V1 V2 V4 - 10 7 9 3 -6 Then Hand K are subspaces of R3. In fact, H and K are planes in R3 through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. W= [Hint: w can be written as C1 V2 + c2V2 and also as c3 V3...
(a). Find the equation of the plane through Po = (1,2,1) with normal vector i = (3,1,2) (b). Find the equation of a plane through Po = (2,3,1) and parallel to the plane P:3x + 2y -- z = 4 | Q4. Consider the line z-3 y-2 3 L, : * - - - L2: **** 2+5 y-3 -1 2 (i). Write the equations of both lines in parametric form (ii). Find the direction vectors V1, V2 of the lines...
2 5 Do the vectors u = and v= 3 7 span R3? -1 1 Explain! Hint: Use Let a, a2,ap be vectors in R", let A = [a1a2..ap The following statements are equivalent. 1. ai,a2,..,a, span R" = # of rows of A. 2. A has a pivot position in every row, that is, rank(A) Select one: Oa. No since rank([uv]) < 2 3=# of rows of the matrix [uv b.Yes since rank([uv]) =2 = # of columns of...
6. Let P be the subspace in R 3 defined by the plane x − 2y + z
= 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal
vectors that form a basis for P. (b) [5 points] Find the projection
p of b = (3, −6, 9) onto P.
6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...
(c) Each equation below specifies a line or a plane in R3. If possible, express the specified line or plane as a span. Otherwise, justify why it cannot be expressed as a span. i. 2x-yz=4 ii. х+6у—z%3D 0 iii. x+3z = 0 iv. у %3D1 v. x = 0 and z = 0 vi. 2x -y 2 and z =-1 (d) For lines or planes in question 4c that cannot be expressed as spans, express as a translated span