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Math Problem: Linear Algebra Assignment Paper #4 (Math Problem Sample)

Instructions:

THE TASK IS AN ASSIGNMENT ON LINEAR ALGEBRA.

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Content:

Linear Algebra: Assignment 4
University Affiliation
Student Name
Course Name
Date
Introduction
Assignment 4 of the linear algebra module deals with elementary vector operations with applications of algebra methods to solve linear systems of equations. Methods, properties and theorems are adopted from Howard Anton & Chris Rorres book, Elementary Linear Algebra (11th Edition). The solutions are herein indicated stepwise and in order of questions as they appear in the original assignment file.
Question 1(a)
u=(1,2,-3,5,0)
v=(0,4,-1,1,2)
w=(7,1,-4,-2,3)
* v+w=0,4,-1,1,2+7,1,-4,-2,3
=0+7,4+,1,-1-4,1-2,2+3
=7,5,-5,-1.5
* 3(2u-v=3(21,2,-3,5,0-(0,4,-1,1,2))
=3(2,4,-6,10,0-(0,4,-1,1,2))
=3(2-0,4-4,-6+1,10-1,0-2)
=3(2,0,-5,9,-2))
=(6,0,-15,27,-6)
* 3u-v)-(2u+4w)=
=31,2,-3,5,0-0,4,-1,1,2-21,2,-3,5,0+7,1,-4,-2,3
=(3,6,-9,15,0-(0,4,-1,1,2))-(2,4,-6,10,0+(28,4,-16,-8,12))
=3-0,6-4,-9+1,15-1,0-2-2+28,4+4,-6-16,10-8,0+12
=3,2,-8,14,-2-30,8,-22,2,12
=(-28,-6,-30,12,-14)
* 12(w-5v+2u)
=127,1,-4,-2,3-50,4,-1,1,2+21,2,-3,5,0+0,4,-1,1,2
=127,1,-4,-2,3-0,20, -5,5,10+2,4,-6,10,0+(0,4-1,1,2)
=127-0+2,1-20+4,-4+5-6,-2-5+10,3-10+0+0,4,-1,1,12
=129,-15,-5,3-7+(0,4,-1,1,2)
=92,-152,-52,32,-72+0,4-1,1,2
=(92,-72,-72,52,-32)
Question 1(b)
Given:
c1-1,0,2+c12,2,-2+c31,-2,1=(-6,12,4)
-c1+2c2+c3=-6 …i
2c2-2c3=12 …ii
2c1-2c2+c3=4 …(iii)
Solve eqn i and eqn (iii) to eliminate the variable c1
-2c1+4c2+2c3=-12
2c1-2c2+c3=4
Adding the two equations, the following resultant equation is obtained:
2c2+3c3=-8 …(iv)
Solving equation (ii) and equation (iv):
2c2-2c3=12
2c2+3c3=-8
⟹5c3=-20
c3=-4
∴ c2-c3=6 ⟹ c2=2
Also:
c1=6+2c2+c3
c1=6+4-4
c1=6
Question 2(a)
* Given:
u=(1,2-3,0)
v=5,1,2,-2
∴d=1-52+2-12+-3-22+(0--21)2
=16+1+25+4
46
and
cosθ=u.vuv=1*5+2*1+-3*2+0*212+22+(-3)2+02*52+12+22+(-2)2
=5+2-6+01434=11434
θ=cos-1(11434)=87.37
Hence θ<900
* u=(0,1,1,1,2)
v=2,1,0,-1,3
∴d=4+0+1+4+1=10
cosθ=0+1+0-1+61+1+1+44+1+1+9=6715
θ=cos-167=54.150
θ is an acute angle since θ<900
Question 2(b)
According to Cauchy-Schwarz inequality
u.v≤uv
* u=(4,1,1)
v=1,2,3
u.v=4*1+1*2+1*3
=9
u.v=9
u=16+1+1=18
v=1+4+9=15
u.v=18*15=16.43

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