Table of Contents

- 1 Can a vector space have only one vector?
- 2 Is a vector space that is contained within another vector space?
- 3 How do you determine if a set of vectors is a vector space?
- 4 Can a vector space have more than one basis justify?
- 5 How many vectors are in a basis?
- 6 Can there be 2 bases?
- 7 Which plane is a subspace of the full vector space?
- 8 What is the difference between R2 and R3 vector space?

## Can a vector space have only one vector?

Yes, x+y does equal y + x because both are 0, the only vector in the space. The same argument applies to verify VS2 and VS5 through VS8. Each axiom is an equation which says one vector equals another vector, but since there’s only one vector, namely 0, both sides have to be that vector.

## Is a vector space that is contained within another vector space?

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space.

**Does every vector space have exactly one basis?**

All the linearly independent vectors of V that spans V are called as the basis of V. Thus, the vector spaces of the form have exactly one basis.

### How do you determine if a set of vectors is a vector space?

To check that ℜℜ is a vector space use the properties of addition of functions and scalar multiplication of functions as in the previous example. ℜ{∗,⋆,#}={f:{∗,⋆,#}→ℜ}. Again, the properties of addition and scalar multiplication of functions show that this is a vector space.

### Can a vector space have more than one basis justify?

(d) A vector space cannot have more than one basis. (e) If a vector space has a finite basis, then the number of vectors in every basis is the same.

**How many vectors are there in the basis for the vector space of 4d vectors?**

4

In other words, we will have a set of 4 linearly independent vectors in a 4-dimensional space–Theorem 4 tells us that this will be a basis.

## How many vectors are in a basis?

So there are exactly n vectors in every basis for Rn . By definition, the four column vectors of A span the column space of A. The third and fourth column vectors are dependent on the first and second, and the first two columns are independent. Therefore, the first two column vectors are the pivot columns.

## Can there be 2 bases?

It is not hard to check that any vector space (over an infinite field) has infinitely many bases. In a trivial way, you could vary the length of the vectors to get a different basis, and of course you can do this in infinitely many ways.

**Can a vector space have more than one zero vector?**

No, it’s not even possible for a vector space to have more than one zero vector. Every vector space has exactly one vector which is neutral for addition, namely, a zero vector. No; it’s like saying can a numberline have two zeroes.

### Which plane is a subspace of the full vector space?

The plane going through .0;0;0/ is a subspace of the full vector space R3. DEFINITION A subspace of a vector space is a set of vectors (including 0) that satisﬁes two requirements: If v and w are vectors in the subspace and c is any scalar, then (i) v Cw is in the subspace and (ii) cv is in the subspace.

### What is the difference between R2 and R3 vector space?

The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane: v D.x;y/. Similarly the vectors in R3 correspond to points .x;y;z/ in three-dimensional space.

**Do all vector spaces have to obey the 8 rules?**

All vector spaces have to obey the eight reasonable rules. A real vector space is a set of “vectors” together with rules for vector addition and multiplication by real numbers. The addition and the multiplication must produce vectors that are in the space.