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Unformatted text preview: Math 136 Assignment 3 Due: Wednesday, Jan 26th 2 l
1. Let {i = 1 and 27: ——1 . Calculate projgﬂ and perpi; 12’.
3 1
l
2. Find the projection of 27 = 1 onto the plane with vector equation
2
l —~2
55:3 —2 +t 1 , 3,tER
l —~2 3. Let f E R“ and c E R. Prove that Has?“ 2 [c[[[f[[. 4. Determine Whether each of the following statements is TRUE or FALSE. Justify each
choice by giving a proof, or a counter—example. a) Let EEER”. Ire—5.5:: “é’forallc'iER”, thené’za b) Let (i be a nonzero vector in R”. The set {projai} perpa if} is linearly independent
for any i" 6 IR”. C) H1227 6 R", with 22' # 5, 17% 6, and {517 0, then the set with vector equation span{11’, 27} is a plane in R”. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems,
and submit written answers to the questions along with the rest of your assignment. Linear Combinations and Properties of Vectors Review the posted MATLAB Introduction before attempting Questions 1 and 2 below. (See the
course webpage in UVV—ACE under Content > MATLAB.) In particular, a review how to enter vectors in MATLAB,
a ﬁgure out how to add vectors together, and
0 ﬁgure out how to calculate a scalar multiple of a vector. dot
To ﬁnd the dot product of two vectors in MATLAB, use the dot command. For example, the dot product of vectors a = (4, 3, ——1) and b = (—2, 5, 3) can be found as follows: >> a = [4; 8; —1]
>> b = [2; 5; 3]
>> dot(a, b) MATLAB returns that the dot product is 4. norm
To ﬁnd the length of a vector in MATLAB, use the norm command. For example, the length of a from the previous example can be found as follows: >> norm(a) MATLAB returns that the length of a is 5.0990. W Consider the set of vectors {01,122, . . . ,07} in R10 below: 111 = (9, ,,,,,,,5,2075——196~1)
02 —~ (0,~2,—1, 0, 3, 3 4, ~2,4, 1)
03 == (4,~2,—1,0,,334 —::531)
v4 :— (7,5,12,5, 5, —3,~1,6, 3,4)
115 —~ (1,0 ,1,0,1,0,0,0,1,1)
'06 (0,4, 2,0, —6, 6, ”3,4, 3—2)
117 — (2,6,8 ,5,——6,——2,8,8,——2,7) Question 1 Enter the vectors 01,112,. . . ,07 into MAT LAB and then ﬁnd the following linear combinations: (a) azvl+vg+vg+v4+v5+vs+v7
(b) 12:504—907 (c) 0:201—303+305+05v7 Question 2 Find the angle, 6, in radians and degrees, between vectors 02 and 03. Hints:
1. Review the formula for cos 6 in the Course Notes, Section 1.3, p. 18. 2. Type help acos and help acosd at the prompt and read the documentation. ...
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 Spring '08
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