Question One

Thursday, 1 November 2018

3:26 PM

Untitled picture.png Machine generated alternative text:
1. 
i) 
Evaluate each of the following integrals. 
a) 11 
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Untitled picture.png Machine generated alternative text:
dc 
2) 
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Untitled picture.png Machine generated alternative text:
ii) Find the general solution of the differential equation 
d2y 
dy 
+ 2 — + 231 ¯ 2'X. 
dX2 
dx 




































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Untitled picture.png Machine generated alternative text:
1 
iii) Let VI = 
-6 
-8 
1 
and v3 — 
2 
1 
-2 
—1 
1 
a) Write v3 as a linear combination of VI and v2. 
b) Does there exist a linear map T : such that 
T(VI) — 
and T(V3) - 
c) What is the relationship between span(vl, v2) and span(vl, v2, v3)? 
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Untitled picture.png Machine generated alternative text:
1 
iii) Let VI = 
-6 
-8 
1 
and v3 — 
2 
1 
-2 
—1 
1 
a) Write v3 as a linear combination of VI and v2. 
b) Does there exist a linear map T : such that 
T(VI) — 
and T(V3) - 
c) What is the relationship between span(vl, v2) and span(vl, v2, v3)? 










































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Untitled picture.png iv) Let S — {x e IR3 : x2 
a) Prove that S is closed under scalar multiplication. 
b) Show that S is not a subspace of 
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