Homework Answers
3. Let ū and ū be vectors. Prove that ū x ū is orthogonal to both...
3. Let ū and ū be vectors. Prove that ū x ū is orthogonal to both ū and v.
4. Let ū, w be vectors. Prove the Parallelogram Law: || + 2011? + || D...
4. Let ū, w be vectors. Prove the Parallelogram Law: || + 2011? + || D – ||2 = 2 || || 2 + 2 ||0||2.
5 3 1 Let ū = < 2,-3> V = <-2,0 > w = <3,3 >...
5 3 1 Let ū = < 2,-3> V = <-2,0 > w = <3,3 > Graph vectors ū, ū, and w in standard position with corresponding terminal points, A, B, and C, respectively. (72 point) What is the length of the altitude of AABC from vertex A? (72 point) -5 -3 -1 -1 0 1 3 5 -3 -5
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle betwe...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Given the following vectors: ū= 3 ū= W = > (a) Find the projection of ū...
Given the following vectors: ū= 3 ū= W = > (a) Find the projection of ū onto ū. BOX YOUR ANSWER. (b) Find the projection matrix of the projection in part (a). BOX YOUR ANSWER. (c) Find the projection of ū onto the subspace V of R3 spanned by ✓ and W. (You may use MATLAB for matrix multiplication in this part, but you must provide the expressions in terms of matrices.) BOX YOUR ANSWER. (d) Find the distance from...
Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written...
Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written in row form, 3 that is, of = [1 2 3]. It can be seen as a 1 x 3 matrix. For every vector R3, set f(w) = 1 WER. (i) Show that f: R3 → R defines a linear transformation. (ii) Show that f(ū) > 0. (iii) What are the vectors we R3 such that f(w) = 0?
linear alegbra Let u, v, w be linearly independent vectors in R3. Which statement is false?...
linear alegbra
Let u, v, w be linearly independent vectors in R3. Which statement is false? (A) The vector u+v+2w is in span(u + u, w). (B) The zero vector is in span(u, v, w) (C) The vectors u, v, w span R3. (D) The vector w is in span(u, v).
(10 points) Are the vectors ū linearly dependent [25 1], ū = [-5 -5 o] and...
(10 points) Are the vectors ū linearly dependent [25 1], ū = [-5 -5 o] and ū = [-5 -3 2] linearly independent? If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, find the only scalars that will make the equation below true. ũ+ ö+ ū = 0.
6. Displacement vectors 7 , ū, V, and ū are given below. In the appropriate diagram,...
6. Displacement vectors 7 , ū, V, and ū are given below. In the appropriate diagram, draw (a) the projection of 7 onto ū (b) the projection of ū onto ū (c) the projection of ū onto ū. ū ū ū 2 f V w
41 and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be...
41
and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
3. Let ū and ū be vectors. Prove that ū x ū is orthogonal to both ū and v.
4. Let ū, w be vectors. Prove the Parallelogram Law: || + 2011? + || D – ||2 = 2 || || 2 + 2 ||0||2.
5 3 1 Let ū = < 2,-3> V = <-2,0 > w = <3,3 > Graph vectors ū, ū, and w in standard position with corresponding terminal points, A, B, and C, respectively. (72 point) What is the length of the altitude of AABC from vertex A? (72 point) -5 -3 -1 -1 0 1 3 5 -3 -5
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Given the following vectors: ū= 3 ū= W = > (a) Find the projection of ū onto ū. BOX YOUR ANSWER. (b) Find the projection matrix of the projection in part (a). BOX YOUR ANSWER. (c) Find the projection of ū onto the subspace V of R3 spanned by ✓ and W. (You may use MATLAB for matrix multiplication in this part, but you must provide the expressions in terms of matrices.) BOX YOUR ANSWER. (d) Find the distance from...
Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written in row form, 3 that is, of = [1 2 3]. It can be seen as a 1 x 3 matrix. For every vector R3, set f(w) = 1 WER. (i) Show that f: R3 → R defines a linear transformation. (ii) Show that f(ū) > 0. (iii) What are the vectors we R3 such that f(w) = 0?
linear alegbra
Let u, v, w be linearly independent vectors in R3. Which statement is false? (A) The vector u+v+2w is in span(u + u, w). (B) The zero vector is in span(u, v, w) (C) The vectors u, v, w span R3. (D) The vector w is in span(u, v).
(10 points) Are the vectors ū linearly dependent [25 1], ū = [-5 -5 o] and ū = [-5 -3 2] linearly independent? If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, find the only scalars that will make the equation below true. ũ+ ö+ ū = 0.
6. Displacement vectors 7 , ū, V, and ū are given below. In the appropriate diagram, draw (a) the projection of 7 onto ū (b) the projection of ū onto ū (c) the projection of ū onto ū. ū ū ū 2 f V w
41
and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
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