# I Is zero vector always present in any n-dimensional space?

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1. Jul 28, 2017

In the book, Introduction to Linear Algebra, Gilbert Strang says that every time we see a space of vectors, the zero vector will be included in it.

I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?

2. Jul 28, 2017

### andrewkirk

A plane in 3D space is not a vector space unless it passes through the origin. It is an Affine Space, which is a generalisation of the concept of a vector space.

The reason that a plane that doesn't pass through the origin is not a vector space is that it is not closed under addition or scalar multiplication. Consider the plane $z=2$, which contains the vectors (0,0,2) and (1,1,2). The sum of those is (1,1,4) which is not in that plane.

3. Jul 29, 2017

Can you kindly explain that how (1,1,2) lies in the plane z=2. The plane z=2 should only contain (0,0,2). No?

4. Jul 29, 2017

### andrewkirk

No. The plane z=2 is the set of all points (x,y,2) where x and y are any real numbers.

5. Jul 29, 2017

### weirdoguy

If it would contain only one point it wouldn't be a plane...

6. Jul 29, 2017

### Staff: Mentor

To expand on what andrewkirk said, the equation z = 2 means that both x and y are completely arbitrary. You are assuming that since x and y aren't mentioned, both must be zero. Instead, since they aren't present, they can take on any values.

7. Jul 31, 2017

### Lineisy Kosenkova

I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?

8. Aug 1, 2017

### Skins

If the plane does not pass through the origin then it is not a vector space (according to definition a vector space must contain the zero vector).