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. Enter your answer as a subset of {1

 

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. Enter your answer as a subset of {1

 

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) = + Enter your answer as a subset of {1

 

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) = + Enter your answer as a subset of {1

 

 

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