What is the difference between zero scalar and zero vector?

[dexterdev]

Can anyone clarify the concepts of zero vector and zero scalar?

-Devanand T

 

[Stephen Tashi]

Are you familiar with reading mathematical definitions? By that, I mean do you interpret them as they are written rather than try to make up something "in your own words"?

The definition of a vectors space says it has a set of scalars and a set of vectors. It doesn't require these to be the same set. For example, we can have a vector space where the scalars are the set of rational numbers and the vectors are the set of all functions defined on the unit interval (defining addition of vectors to be the usual addition of functions). The zero of the scalars is the number zero. The zero of the vectors is the function defined by f(x) = 0 for each x in the unit interval. As you recall, a function is a special kind of set of ordered pairs of numbers. The number zero is a single number.

Another interesting observation is the non-universality of the meaning of "=". To say two scalars are equal doesn't mean the same thing as saying that two vectors are equal. If you think of the test for equality being implemented by a computer algorithm, testing the equality of two numbers is a different algorithm that testing the equality of two functions.

As to comparing scalars with vectors, the definition of vector space does not say that there is any "=" relation defined between a scalar and a vector. So technically you can't say a scalar is "not equal" to a vector either! You should simply say that no equality relation is defined (in the definition of a vector space) between scalars and vectors.

The definition of a vector space tells about properties that a vector space must have. It doesn't prohibit the vector space from having additional properties. Does it prohibit you from making an example where the scalars and vectors are the same set? I suppose we'd have to read the definition carefully to see. I don't think it does.

 

[HallsofIvy]

One is a scalar and the other is a vector? You can add two vectors or add two scalars but you cannot add a vector and a scalar. If a and 0 are two scalars, then a+ 0= a. If \vec{v} and \vec{0} are vectors, then \vec{v}+ \vec{0}= \vec{v}. But neither a+ \vec{0} nor \vec{v}+ 0 are defined.

On a more practical note, in a finite dimensional vector space, of dimension n, say, with a given basis, we can represent any vector as a linear array of scalars: < a, b, c, ...>. The 0 vector would be represented as <0, 0, 0, ...> while the scalar 0 is just the single scalar.

In a function space, that might be of infinite dimension, the 0 vector is the function f(x)= 0 that is 0 for all x while, again, the 0 scalar is a single number.

 

[Fredrik]

If you're dealing with a vector space over $\mathbb R$ (i.e. if $\mathbb R$ is the set whose members will be called "scalars"), then the simple answer is that the 0 scalar is a member of $\mathbb R$, and the 0 vector is a member of the vector space.

$\mathbb R$ can actually be thought of as a vector space over $\mathbb R$. When we're dealing with that specific vector space, the 0 scalar and the 0 vector are the same.