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MAT 216 Sample Problems

This document provides 7 sample problems related to mathematics: (1) showing an injective map between a unit cube and unit segment, (2) proving or disproving a statement about a norm on R^2, (3) showing Cauchyness of a sequence of determinants, (4) proving existence of an integer n such that image and kernel of a linear map intersect trivially, (5) describing boundaries of open and closed sets, (6) proving intersection of closed cells is a single point, (7) showing absolute convergence of a power series.

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216 views1 page

MAT 216 Sample Problems

This document provides 7 sample problems related to mathematics: (1) showing an injective map between a unit cube and unit segment, (2) proving or disproving a statement about a norm on R^2, (3) showing Cauchyness of a sequence of determinants, (4) proving existence of an integer n such that image and kernel of a linear map intersect trivially, (5) describing boundaries of open and closed sets, (6) proving intersection of closed cells is a single point, (7) showing absolute convergence of a power series.

Uploaded by

david.contatos4308
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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MAT 216 Sample Problems

1) Show that the there exists an injective map from the unit cube (0, 1)3 into the unit
segment (0, 1). Is there a bijection between these two sets?

2) Consider the norm on R2 given by kxk1 = |x1 | + |x2 |. Prove or disprove the following
statement: there exists an inner product h·, ·i : R2 p
× R2 → R which makes the vector space
2
R into an inner product space, such that kxk1 = hx, xi.

3) Let d2 (A, B) be the Euclidean distance on the space of real-valued n × n matrices. Show
that if the sequence of matrices {An }n≥1 ⊂ Rn×n is Cauchy, then the sequence of their
determinants {det(An )}n≥1 is Cauchy in R.

4) Show that if T : V → V is a linear mapping, where V is a finite-dimensional vector space,


there exists an integer n for which im(T n ) ∩ ker(T n ) = 0.

5) In a topological space (S, T ), what is the boundary of a set which is both open and
closed? For S = [0, 1] with the topology induced by the absolute value metric, is there a set
C whose boundary ∂C is uncountable?

6) Show that if {∆ ¯ n }n≥1 are a nested sequence of closed cells in Rd , i.e. with ∆
¯ n+1 ⊆ ∆
¯ n,
¯
such that their diameters en = diam(∆n ) vanish as n → ∞, then the their intersection
¯ n consists of a single point in Rd .
∩n≥1 ∆

7) Let α > 0, and consider the power series


X α(α − 1) . . . (α − n + 1)
xn .
n≥0
n!

Show that this power series converges absolutely for all x ∈ [−1, 1]. Is the resulting function
continuous on [−1, 1]?

8) Consider the space of absolutely summable real sequences


( )
X
`1 (R) = {an }n≥1 : an ∈ R, k{an }k1 := |an | < ∞ .
n≥1

Show that this is a complete metric space with respect to the P distance induced by the norm
k · k1 , i.e. with respect to d1 ({an }, {bn }) = k{an − bn }k1 = n≥1 |an − bn |.

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