Alg1231W2T3 - Examples and counterexamples

Wednesday, 25 July 2018

9:10 PM

Machine generated alternative text:
Here are the essential features that make R into a vector space [select all that apply]: 
We have a well-defined notion of addition of vectors 
We have a well-defined notion of multiplication of vectors 
We have a well-defined notion of division of vectors 
We have a well-defined notion of multiplying a vector by a scalar 
These operations satisfy a particular list of properties 
The basic list of required properties includes [select all that apply]: 
If u and v are vectors then u + v = v u 
Cl If u and v are vectors then u x v = v x u 
If u, v and w are vectors then (u + v) + w = u + (v + w) 
Cl If u, v and w are vectors then (u — v) — w = u — (v — w) 
Cl There is a special vector 1 with the property that 1 x v 
v for any vector v 
There is a special vector O with the property that O + v 
v 0 = v for any vector v 
If and are scalars and v is a vector then ( Ag)v 
If and are scalars and v is a vector then (A + = + 
Note: this is not a comprehensive list, for that see page 4-6 of your yellow notes 
In fact all of these statements apply not just to making R into a vector space, they also apply to: [select all that apply] 
the vectors v — 
the matrices A 
y , making R into a vector space 
all (112 
making M22 into a vector space 
(121 (122 
41 the polynomials p(x) 
ao + al x + a,-2X , making P2 (IR) into a vector space.

 

Machine generated alternative text:
A subset S of IR2 can often be visualised geometrically A subset S of IR2 is a subspace if it is also a vector space on its own; using the same operations of vector addition and scalar 
multiplication as in R . Most subsets of R are not subspaces. Subspaces are very special and easy to identify geometrically 
Match up the following sets with the appropriate diagrams 
—1 
T 
R 
s 
v 
w 
velR 
velR 
veR 
y 
y 
y 
with 
with 
with 
with 
with 
-3 
y 
2m 
-2 
and 
2 
2 
2 
3 
-3 
-2 
2 
3

 

Machine generated alternative text:
Which of the subsets above is also a subspace of R ?

 

Machine generated alternative text:
Consider the set 
R 
Which of the following vector space properties hold for R [select all that apply] 
41 1. Closure under Addition. If u, v e R, then u + v e R 
velR 
with 
y 
41 2. Associative Law of Addition. If u, v, w e R: then (u + v) + w = u + (v + w). 
41 3. Commutative Law of Addition. If u, v e R, then u + v = v + u. 
41 4. Existence of Zero. There exists an element O e R such that for all v e R, v +0 = v. 
Cl 5. Existence of Negative. For each v e R there exists an element w (usually denoted —v) such that v + w 
Cl 6. Closure under Multiplication by a Scalar. If v e R and e IR, then e R. 
41 7. Associative Law of Multiplication by a Scalar. If X," e IR and v e R; then A(gv) = 
8. Scalar Multiplication Identity. If v e V, then Iv = v. 
9. Scalar Distributive Law. If e IR and v e R, then (A + 'L)v = .Xv + 
41 10. Vector Distributive Law. If e IR and u, v e R, then + v) = Au + Av. 
Here is a proof that R does not satisfy closure under multiplication by a scalar. 
1 
Consider the scalar 
—1 and the vector v 
, which belongs to R since its first component is x 
1 > O. Then 
—1 
is not in R since its first component is 
Thus R is not closed under multiplication by a scalar. 
Note that to prove a condition is not satisfied we only need to exhibit a single counter-example Be careful about how you try to construct your counter-example though, if we had chosen v 
or had chosen 
then we would not have produced a counter-example.

 

Machine generated alternative text:
The previous question's discussion of counter-examples is worth amplifying. To check that a subset of S is a subspace of a vector space V, it is necessary that each of the properties of a vector 
space hold for S However to show that S is not a subspace we only need to find one example of a single property which does not hold. 
Which of these "real life" examples are analogous: [select all that apply] 
A car is top notch if it always starts 
A car is not top notch if it sometimes and sometimes doesn't 
Cl Saying a car is not top notch means that it never starts. 
To prove that a computer program has a bug, we only need to find one example where it does not work properly 
Cl To prove that a computer program has a bug, we need show that it never works properly 
To prove that your love is true, he/she ought to be faithful every day 
Cl To prove that your love is true, she/he only needs to be faithful once. 
Cl To prove that your love is untrue, she/he needs to be unfaithful every day 
To prove that your love is untrue, she/he needs only to be unfaithful once. 
Okay maybe we are pushing the analogy here, but we hope you get the idea

 

Machine generated alternative text:
Let's check your understanding: Select the statements below that can be proven by finding a suitable example or counter-example, and those which require a more general proof. 
The question (or statment): 
Given a convex quadrilateral ABCD; prove that the 
uadrilateral formed by joining the midpoints of 
AB, BC, CD, and DA is a parallelogram. 
Is the set {O, 1, 2, 3, 4, 5, 6, 7, 8, 9} closed under 
dditlon? Prove your answer. 
Show that, in general; ATA # AAT even if A is 
uare_ 
Prove that llal — lbll la — bl for all a, b e IRn 
10 
here exists 10 vectors in R , each pair of which is 5 
part 
here exist a real matrix G such that 
GCT _ 45 0 
0 20 
Suppose that v and w are vectors in IR3- and are 
real numbers Prove the scalar distributive law 
(A + = + uv and the vector distributive law 
can be answered 
(or proven) using a: 
general proof 
suitable example 
suitable example 
general proof 
suitable example 
suitable example 
general proof

 

Machine generated alternative text:
Show that, in general; matrices cannot be cancelled from 
products (i.e. show that AB = AC does not necessarily 
imply B = C). 
Let Q be a square n x n orthogonal matrix. Show that 
e columns of Q are a set of n orthonormal vectors in 
Show that, in general; det(AA) # det(A). 
Show that, in general; a square orthogonal matrix Q has a 
alue for det(Q) of +1 or —1 _ 
suitable example ' 
general proof 
suitable example 
general proof

 

 

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