Alg1231W2T1 - Vector space properties

Tuesday, 24 July 2018

10:17 PM

Machine generated alternative text:
Suppose that 
2 
Define addition of vectors and scalar multiplication in the usual way as in R 
Which of the following 10 properties of a vector space are satisfied by T ? 
Closure under Addition. If u, v e T then u + v e T 
2 Associative Law of Addition. If u, v, w e T , then (u + v) + w = v + (u + w). 
3 Commutative Law of Addition. If u, v e T then u + v = v + u. 
4 Existence of Zero. There exists an element O e T such that, for all v e T , v + O 
and 
y 20 
5 Existence of Negative. For each v e T there exists an element w (usually denoted —v ) such that v + w 
6 Closure under Multiplication by a Scalar. If v e T and e R then Av e T. 
7 Associative Law of Multiplication by a Scalar. If e R and v e T , then A(gv) = (Ag)v. 
8 Scalar Multiplication Identity. If v e V , then Iv = v. 
g Scalar Distributive Law. If A, g e IR and v e T , then (X + = + "v. 
ID Vector Distributive Law. If e IR and u, v e T , then + v) = + Av. 
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Machine generated alternative text:
Suppose that 
S 
is the set of quadratic functions which have graphs opening upwards 
Define addition of functions by the usual rule; so that for example 
ax2 -F bx+c : a, e R, a > O 
2 
— + I) + = + + 18, 
and define scalar multiplication also by the usual rule, so that for example 
Which of the following 10 properties of a vector space are satisfied? 
Closure under Addition. If f,g e S, then f + g e S. 
2 Associative Law of Addition. If f, g,h e S, then (f + g) + h = f + (g + h). 
3 Commutative Law of Addition. If e S, then f g = g + f. 
4 Existence of Zero. There exists an element 0 e S such that; for all f e S , f +0 = f. 
5 Existence of Negative. For each f e S there exist an element h e S (usually written as —f ) such that f + h 
6 Closure under Multiplication by a Scalar. If f e S and e IR , then A f e S. 
7 Associative Law of Multiplication by a Scalar. If A, ß e R and f e S then A(gf) = (Ag)f. 
8 Scalar Multiplication Identity. If f e S then If = f. 
g Scalar Distributive Law. If e R and f e S then (A + = Af + 'If. 
ID Vector Distributive Law. If e R and e S , then + g) — X f + Ag. 
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Machine generated alternative text:
We will prove that v + v = 2v, for every vector v in a vector space V. 
Using the property & scalar Multiplication Identity 
Then property 9. scalar Distributive Law 
O allows us to write down the identity 
O allows us to rewrite this as 
And since, by definition, 1 + 1 = 2 in any field F, we get that v + v = 2v as required. 
Vector Space Properties: 
Closure under Addition. If u, v V then u + v e V. 
2 Associative Law of Addition. If u, v, w e V , then (u + v) w = v + (u w). 
3 Commutative Law of Addition. If u, v e V then u + v = v + u. 
4 Existence of Zero. There exists an element O e V" such that, for all v e V , v +0 = v. 
5 Existence of Negative. For each v e V there exists an element w (usually denoted —v ) such that v + w 
6 Closure under Multiplication by a Scalar. If v e V and e F , then e V. 
7 Associative Law of Multiplication by a Scalar. If and v e V then A(gv) = (Ag)v. 
8 Scalar Multiplication Identity. If v e V, then IV = v. 
g Scalar Distributive Law. If F and v V then (A + = hr 4- 'tv. 
ID Vector Distributive Law. If e F and u, v e V , then + v) = Au + Av.

 

The 13th century Italian mathematician Leonardo of Pisa (pictured) was also known as Fibonacci 
LEONARDO 
I'ISANO 
Sometimes we like to consider sequences a = (al , a,2, (13, • • where ai e Q, just like vectors Now consider the set of Fibonacci-type sequences, that is 
: where an +2 = an+l -Fan for n 2 1}. 
Define vector/sequence addition and scalar multiplication (over Q) of elements of F component-wise; so that for example 
= (3, 5,8, 
26, • • 
and 
= (3, 3, 6, 9, 15, 24, • 
Which of the following 10 properties of a vector space are satisfied? 
1 Closure under Addition. If a, b e F, then a + b e F. 
2 Associative Law of Addition. If a, b,c e F; then (a + b) + c = a + (b + c). 
3 Commutative Law of Addition. If a, b e F, then a + b = b + a. 
4 Existence of Zero. There exists an element O e F such that, for all a e F, a + O = a. 
5 Existence of Negative. For each a e F there exists an element c (usually denoted —a) such that a + c 
6 Closure under Multiplication by a Scalar. If a e F and e Q, then e F. 
7 Associative Law of Multiplication by a Scalar. If e Q and a e F; then A(ga) = (Ag)a. 
8 Scalar Multiplication Identity. If a e V, then la = a. 
g Scalar Distributive Law. If e Q and a e F, then (A + = Aa + pa. 
ID Vector Distributive Law. If e Q and a, b e F, then Ma + b) = + 
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Suppose that 
Define addition + on C by the unusual rule 
C = {f@) : f is a real-valued continuous function on [0, 1]} . 
max (f(x), g(x)) , 
for f,geC and x e [0, 1]. 
This addition comes from the max-plus algebra and is of practical use in control theory 
Define scalar multiplication by the usual rule 
Af@), for f e C, x e IR and x e [0, 1] . 
Which of the following 10 properties of a vector space are satisfied? 
1 Closure under Addition. If e C , then fog e C. 
2 Associative Law of Addition. If f,g,h e C then (f g) h = f + (g + h). 
3 Commutative Law of Addition. If e C , then f G g = g + f• 
4 Existence of Zero. There exists an element 0 e C such that, for all f e C , f 60 = f. 
5 Existence of Negative. For each f e C there exist an element h e C (usually written as —f ) such that f + h 
6 Closure under Multiplication by a Scalar. If f e C and e R then X f e C. 
7 Associative Law of Multiplication by a Scalar. If A, ß e R and f e C then A(gf) — (Ag)f. 
8 Scalar Multiplication Identity. If f e C, then If = f. 
9 Scalar Distributive Law. If A, g e IR and f e C , then (A + = gf. 
10 Vector Distributive Law. If e IR and e C then g) = + 
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