Is the zero vector space v={0}, linearly independent. If so, why is that?
To determine whether the zero vector space v={0} is linearly independent, we need to understand the definition of linear independence.
A set of vectors is linearly independent if none of its vectors can be written as a linear combination of the others. In other words, if the equation \(\alpha \mathbf{v} = \mathbf{0}\) has only the trivial solution, where \(\mathbf{v}\) is a vector in the set and \(\alpha\) is a scalar, then the set is linearly independent.
In the case of the zero vector space v={0}, there is only one vector, which is the zero vector itself. Therefore, we cannot find any non-zero vector \(\mathbf{v}\) in the set that can be written as a linear combination of the zero vector.
Let's consider the equation \(\alpha \mathbf{0} = \mathbf{0}\) for any scalar \(\alpha\). It is clear that the equation will always be satisfied because multiplying any scalar by the zero vector still gives the zero vector.
However, this equation only has the trivial solution \(\alpha = 0\). If we try to find a non-zero scalar \(\alpha\) that satisfies the equation, we will not be able to find one.
Therefore, the zero vector space v={0} is linearly independent, as there is no non-zero vector that can be written as a linear combination of the zero vector.