# 2 simple vector spaces question

In summary, the conversation discusses two questions about vector spaces. The first question, regarding the set of all n-tuples of real numbers with the standard operation on R^n, does not make sense as the operations on R^2 cannot be applied to R^n. The second question, regarding the set of all positive real numbers with operations x+y = x*y and kx = x^2, asks if it is a vector space. The expert suggests checking the axioms or rules to verify if it is a vector space. Additionally, the expert mentions that there are a few more rules to check, such as the distributive property.
hi,i got 2 question about vector spaces :

1. Do the set of all n-tuples of real numbers of the form (x, x1 ,x2...xn) with the standard operation on R^2 are vector spaces?

2.Do the set of all positive real numbers with operations

x+y =x*y and kx=x^2 are vector space?

1 makes no sense. What do "the standard operations of R^2", which are undefined, have to do with the n-tuples of real numbers? R^n is a vector space.

2, just try and verify the axioms, or figure out where they may go wrong.

those question is from a book ...

And does the book use phrases like "those question"?

Once again: your first question makes no sense. It would make sense if you asked "Does the set of all n-tuples of real numbers of the form (x1 ,x2...xn) with the standard operation on R^n form a vector space?". In that case the anser is obviously "yes". It makes no sense to talk about " real numbers of the form (x1 ,x2...xn) with the standard operation on R^2" because you can't apply the operations on R^2 to R^n.

The second question, "2.Is the set of all positive real numbers with operations
x+y =x*y and kx=x^2 a vector space?" is reasonable. Matt Grimes' point was that it is just a matter of checking the axioms for (or definition of) a vector space.

so you are saying that :
the set of all positive real numbers with operations
x+y =x*y and kx=x^2 are vector space?

No, we are saying you should check to see if the "axioms" for a vector space are satisfied yourself. In particular, is the "distributive law", k(x+y)= kx+ ky, satisfied?

( a 1 ) --- is this 2x2 matrix a vector space ? sorry for asking this coz this is abstract
1 b me

Why is it abstract? What are the operations, for instance, what is k*M for some scalar k? Is that in the set? Check the rules, there really is nothing complicated or hidden in this:

to check if S is a vector space over R, say, check the rules: is ks in S when s is in S and k is in R? ie if s satisfies the rules to be in S, does ks? Similarly what about s+t when s and t are in S? is there a zero vector? Just three simple rules to check.

there are a few more rules to check too (like the afformentioned distributive property).

## 1. What are vector spaces?

Vector spaces are mathematical structures that consist of a set of vectors and a set of operations, such as addition and scalar multiplication, that can be performed on these vectors. They are used to study and solve problems related to linear transformations and systems of linear equations.

## 2. What is the difference between a vector and a scalar?

Vectors are quantities that have both magnitude and direction, while scalars only have magnitude. In other words, vectors can be represented by arrows, while scalars are represented by numbers.

## 3. How are vector spaces defined?

Vector spaces are defined by a set of axioms, or rules, that must be satisfied. These axioms include closure under addition and scalar multiplication, associativity and commutativity of addition, and the existence of an additive identity and inverse for each vector.

## 4. What are some examples of vector spaces?

Some common examples of vector spaces include the set of 2-dimensional vectors in the xy-plane, the set of n-dimensional vectors, and the set of polynomials with real coefficients. Other examples include the set of matrices, the set of functions, and the set of solutions to a homogeneous linear differential equation.

## 5. Can two vector spaces be isomorphic?

Yes, two vector spaces can be isomorphic, meaning that they have the same structure and can be mapped onto each other. This essentially means that they are equivalent, even though they may have different representations and elements.

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