Let ℝ^+denote the set of positive real numbers. Define the "sum" of two elements of 𝐑^+to be the

Let ℝ^+denote the set of positive real numbers. Define the...

Key Concepts

Vector Space Axioms

Vector space axioms are the set of rules that define how addition and scalar multiplication behave within the space. These include properties such as closure under addition and scalar multiplication, commutativity and associativity of addition, existence of a zero vector and additive inverses, and the distributive, associative, and identity properties related to scalar multiplication. Verifying these axioms ensures that the structure qualifies as a vector space over a field, in this case, the field of real numbers.

Isomorphism via Logarithms

An isomorphism is a bijective mapping between two algebraic structures that preserves the relevant operations. For the structure given in the problem, the logarithm function serves as an isomorphism between the positive real numbers (with multiplication as the addition operation) and the real numbers (with the usual addition). This isomorphism simplifies the verification of the vector space axioms by transforming non-standard operations into their standard equivalents.

Vector Space

A vector space is a collection of objects, called vectors, together with operations of addition and scalar multiplication that satisfy a list of axioms such as commutativity, associativity, existence of additive identities and inverses, and distributive properties. This concept is fundamental in linear algebra and geometry, providing a framework for analyzing linear structures in mathematics.

Transcript

00:01 All right, so we are saying that our vector set is the set of all positive real numbers.

00:13 And the addition and scalar multiplication are redefined.

00:23 So now addition on this vector set is defined as.

00:37 Multiplication in the real normal number system.

00:46 And scalar multiplication is redefined as the exponent of the number.

01:02 So is this closed? well, let's just go in order.

01:06 Let's try for or does it meet all of the axioms in order to be a vector space? that's what i meant to say.

01:15 So let's look at axiom one.

01:18 Axiom one states for closure under addition.

01:22 So if i take, and i just realized i didn't write this correctly up here, there we go, if i take x1 plus x2, just some general x1 and some general x2, that's going to equal x1 times.

01:59 X2.

02:00 If this is a real number, positive number, and this is a positive number, or positive times a positive makes a positive.

02:14 And it can't be zero because the only way to get zero is to multiply by zero.

02:21 So this is satisfied.

02:30 A2.

02:34 Closure under scalar multiplication.

02:37 So if i take a constant times a positive real number where c does not have to be a positive real number, i'm trying to look at this to see exactly how c is defined.

02:58 Let me read it again, but i think c can be any real number.

03:03 Note that multiplication and exponentation appearing on the right side of these formulas refer to the ordinary operations of real number.

03:11 On real numbers.

03:15 Okay, so c, or in this case, k, can be positive or negative or zero.

03:24 So this is going to give me x sub 1 to the k power.

03:31 Well, x sub 1 to the k power, if k is greater than 1, is a real number.

03:45 And even if it's less than 1, by greater than zero, it ends up including a radical, but since x sub 1 is positive, that's still a real number.

04:00 So if k is zero, well, then x1 to the zero power is one, which is still a positive real number.

04:10 So what if k is negative? if k is negative, then this ends up with a fraction, would be one over, x1 to the k power, and that's still a real number.

04:24 So we're still good, and that is satisfied.

04:31 But we need to satisfy all 10 conditions.

04:36 So let's go to condition 3, axiom 3, commutativity of addition.

04:45 So what that's telling us is that x1 plus x sub 2 have to equal x sub 2 plus x sub 1.

05:00 Well, x1 plus x2 is defined as x sub 1 times x sub 2.

05:11 But since these are just real numbers, i can reverse the order x sub 2 times x sub 1.

05:17 And so back into our potential vector space, that's x sub 1 plus x sub 2.

05:28 And so, no, no, no, it's x sub 2 plus x sub 1.

05:33 I'm not sure how that happened.

05:38 And so, yes, that is satisfied.

05:41 So let's go down to axiom 4.

05:50 Associativity of addition.

05:52 So that's telling us that x1 plus x2 plus x3, and we have to be able to do 2 plus 3 first and then 1.

06:06 So let's do x1 plus x2 first.

06:09 That would be x1, x2 plus x3, which is going to give us x1, x2, x2, x3.

06:30 Now, according to the associative property, we can rearrange that to x2, x3, x1, which now we can separate to x, whoops, i tried to do green, x2, x3 plus x1, which is x2x3 plus x1, which is x2, x2 plus x3.

07:17 Oh, i mixed up my colors.

07:30 Which is what we wanted.

07:32 So we are good again.

07:34 That axiom is satisfied.

07:37 X, a4.

07:49 Okay, a5.

08:05 Existence of a zero vector.

08:07 I'm wondering about this one.

08:09 Existence of a zero vector.

08:13 So, these are positive numbers.

08:18 So i can't include zero.

08:27 So i'm thinking the only other reasonable candidates are either one or maybe infinity, but infinity is not even a real number.