# Evaluating Vector Spaces: V = {(0,1), (1,0)}

• Kosh11
In summary, the conversation discusses whether the given set V = {(0, 1), (1, 0)} is a vector space over \Re, with the usual notions of addition and scalar multiplication. There is confusion about whether the set is a subset of R^2 or if it includes all vectors in the lines drawn from the origin to (0, 1) and (1, 0). The conclusion is that the set is not a vector space as it only contains the two given vectors and not the zero vector.

## Homework Statement

Are the following vector spaces over $$\Re$$, with the usual notion of
addition and scalar multiplication: V = {(0, 1), (1, 0)}

## Homework Equations

definition of vector space

## The Attempt at a Solution

I'm a little confused by what this means. Am I correcting in thinking in drawing a line from the origin to (0,1) and another from the origin to (1,0) and all the vectors contained in those two lines belong to V? Then I'm unsure as to whether or nor this a vector space. Addition and scalar multiplication seem to hold, but I'm not sure and I'm lost on how to prove it either true or false.

as its written, i would read that as the subset of R^2, with two elements, the vectors ( 0,1) and (1,0), which is clearly not a vector subspace

maybe you should write the question just as it is aksed for clarity

I would interpret that exactly as it is given- as set notation- that (0, 1) and (1, 0) are the only objects in the set. Nothing is said about a "span" or combinations of those vectors. In particular, the 0 vector, (0, 0), is not in the set.

## 1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and two operations, vector addition and scalar multiplication, that satisfy certain properties. These properties include closure, associativity, commutativity, identity element, and inverse element.

## 2. How do you evaluate a vector space?

To evaluate a vector space, you need to check if the set of vectors satisfies the properties of a vector space. These properties include closure, associativity, commutativity, identity element, and inverse element. In this case, we can evaluate the vector space V = {(0,1), (1,0)} by checking if the two vectors satisfy these properties.

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

A vector is a mathematical object that has both magnitude and direction, while a scalar is a mathematical object that only has magnitude. In other words, a vector can be represented by an arrow, while a scalar can be represented by a single number.

## 4. Can a vector space have more than two vectors?

Yes, a vector space can have any number of vectors as long as they satisfy the properties of a vector space. In this case, the vector space V = {(0,1), (1,0)} has two vectors, but there can be vector spaces with three, four, or even more vectors.

## 5. What are some real-life applications of vector spaces?

Vector spaces have numerous real-life applications in fields such as physics, engineering, and computer science. For example, in physics, vectors are used to represent forces, velocities, and other physical quantities. In engineering, vectors are used to represent forces, displacements, and other properties of structures. In computer science, vectors are used in machine learning algorithms to represent data and perform calculations.