Indian Institute of Space Science and Technology
MA121-Vector Calculus Assignment-I
2018
Prosenjit Das
Let Vbe a vector space over R.
1. Show that additive identity of V /Ris unique.
2. Let v∈V. Show that vhas unique additive inverse; and further show that the additive inverse
of vis −1·v
3. Let x1, x2∈Rand v∈V. Suppose, x16=x2. Show that x1·v6=x2·v.
4. Show that V/Rcontains either exactly one element or infinitely many elements.
5. Suppose V=R2and v= (x, y)∈V. Show that vcan not generate V.
6. Suppose V=R3and v1= (x1, y2), v2= (x2, y2)∈V. Show that v1, v2can not generate V.
7. Let V=R2and S={v1, v2} ⊂ V. When will Sgenerate V? Give example of a case where S
will not generate V.
8. Let V=R3and S={(1,1,1),(1,2,3),(1,0,0)}. Show that Sis a basis of V.
9. Let Sbe a basis of V=R2. Show that Scontains exactly two elements.
10. Let S={v1, v2, v3, v4} ⊂ V=R3. Prove that Sis never a basis of V.
11. Let ||.|| be the Euclidean norm on V=R2. Suppose v1, v2∈V. Show that ||v1+v2|| ≤ ||v1||+||v2||.
12. Let < . , . > be the Euclidean inner product ( dot product) on R2. Show the Cauchy-Schwartz
inequality: |< v1, v2>| ≤ ||v1||||v2||. When the equality holds?
13. •If α∈Rbe such that α≥0 and α < 1/n for each n∈N, then show that α= 0.
•Let (a, b) be an open interval, and α, β ∈(a, b) such that α < β. Show that (α, β)⊂(a, b).
•Let A⊂Rbe a bounded set. Define diam(A) := sup{|x−y| | x, y ∈A}. Show that
diam((a, b)) = b−a=diam([a, b]).
•Let (ai, bi) are non-trivial (not-null) open intervals for i∈N. Let A=\
i≥1
(ai, bi). Suppose
that α, β ∈Aand α < β, then show that (α, β)⊂A. What can be said about diam(A)?
•Suppose dn=diam((an, bn)) and {dn} −→ 0, then show that diam(A) = 0 where A=
\
i≥1
(ai, bi); and therefore show that there does not exist α, β ∈Asuch that α < β; and
conclude that A, in this case, is singleton, if each (ai, bi) are non-trivial.
14. Show that in Rn
•both Φ and Rnare open as well as closed sets.
•infinite union of open sets is open; and therefore infinite intersection of closed set is closed.
•finite intersection of open sets is open; and therefore finite union of closed sets is closed.
•infinite intersection of open sets may not be open; and therefore infinite union of closed sets
may not be closed.
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