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Unformatted text preview: Math 136 Assignment 2 Due: Wednesday, Jan 18th 1. Determine, with proof, which of the following are subspaces of R3 and which are not. 3:1 $1
a)81= 0 ERBlZL'1'—$2=O b)32= $2 ER3$1,JE2,$3EZ
£82 LE3
0 $1
C)S3= 0 (1)34: 1'2 ER31$1+$2+IE3=1
0 2'33
2. Let 171, . . . ,6} be vectors in R”. Prove that span{171, . . . ,ﬂk} is a subspace of 1R".
{1’51 2 1
3. Given that S’ = :62 l .731 — 2232 == 503 is asubspace oflR3, prove that B = 1 , 0
LE3 0 1
is a basis for S.
1 2
4. Let "J = 3 and 27 = —1 . Evaluate the following.
——1 1
(a) 1717.
(b) 11’ x 27.
(0) Hill
(d) H17 X 17H 5 Find a scalar equation for the plane with vector equation 1 2 1
:13: 0 +8 3 +t 2 , s,tER
1 3 1 6. Let 171,172 6 Prove + 172H2 2 +H’172H2, then 171 ' 172 2 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 DZL under Content ——> MATLAB.) In particular, a review how to enter vectors in MATLAB,
« ﬁgure out how to add vectors together, and
a 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; 3; ~11
>> b = [2; 5; 3]
>> dotCa, b) MATLAB returns that the dot product is 4. DOI'III
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. Consider the set of vectors {01,212, . . . ,v7} in R10 below: U1 ” (“4: “57276)97ﬂ47372)
v2 (~2,1,0,3,—3,1,—1,1,3,2)
713 : (17“472387—“37773797
v4 = (3,5,4,1,—2,3,—7,8,0,9)
v5 = (0,1,1,0,~1,0?0,0,0,—1)
U6 (4: “2:07ﬂ6a6vﬂ2727*2>67“4)
v7 “ (1,—1,3,—3,0,1,1,~3,3,0)
Question 1
Enter the vectors 111,112, . . . ,v7 into MATLAB and then find the following linear combinations: (a)a=v1+vg+v3+v4+v5+v5+v7 b=6U2*5’U7
(c) c: ——v1+5v2+3v4——4v5+7v6 Question 2
Find the angle, 6, in radians and degrees, between vectors v2 and v5. Hints:
1. Review the formula for cos 0 in the Course Notes, Section 1.3, p. 19. 2. Type help aces and help acosd at the prompt and read the documentation. ...
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