# A question about (0,0,0) vector

• transgalactic
In summary, the dimension of the kernel is the number of vectors in a basis for the null space, which is the set of all vectors mapped to the zero vector by the matrix. If the only vector in the null space is (0,0,0), the dimension of the kernel is 0. This means that the rank of the matrix is equal to the number of columns, and the nullity is 0. For a full-rank matrix, the null space is only the zero vector and its dimension is 0.
transgalactic
when i find that the only vector of ker is (0,0,0)

what is the dim(ker)

is it 1
or is it 0??

does (0,0,0)
counts as a vector

or in case if we find two vectors and one of them is (0,0,0)

what is the dimention of ker now ??

Last edited:
Let A be a matrix. By the rank-nullity theorem, rank A + nullity A = rank A + dim(ker A) = number of columns of A, say n(A). So dim(ker A) = n(A) - rank A.

What is the null space of a full-rank matrix? What is its rank?

Yes, (0,0,0) is a vector. In fact, since adding (0,0,0) to itself gives, again, (0,0,0) and multiplying (0,0,0) by any number gives, again, (0,0,0), (0,0,0) is the only vector in which the set containing only it is a subspace! Since any vector in a basis must be non-zero, a "basis" for {(0,0,0)} must be empty: its dimension is 0.

## What is a (0,0,0) vector?

A (0,0,0) vector is a vector that has a magnitude of 0 and no direction. This means that it has no length and does not point in any specific direction in 3-dimensional space.

## What is the significance of a (0,0,0) vector?

A (0,0,0) vector is significant because it is the only vector that does not have a direction or magnitude. It is often used as a reference point or origin in mathematical calculations.

## Can a (0,0,0) vector be represented graphically?

No, a (0,0,0) vector cannot be represented graphically because it has no length or direction. It would essentially be a point at the center of a 3-dimensional coordinate system.

## What are some real-world applications of a (0,0,0) vector?

A (0,0,0) vector is commonly used in physics and engineering to represent a point of origin or reference in a system. It can also be used in computer graphics to represent the center of a 3-dimensional object.

## Can a (0,0,0) vector have any other values besides 0?

No, a (0,0,0) vector can only have values of 0 for all three components. Any other values would result in a vector with a magnitude and direction, which would no longer be a (0,0,0) vector.

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