Why is the dot product of a vector with itself not a linear function? If an inner product is linear by definition, i.e., $\langle\mathbf{v+w},\mathbf{u}\rangle=\langle\mathbf{v},\mathbf{u}\rangle+\langle\mathbf{w},\mathbf{u}\rangle
$ and $\langle a\mathbf{v},\mathbf{w}\rangle=a\langle\mathbf{v},\mathbf{w}\rangle$, and the dot product is an inner product, then why is $f(\mathbf{x})=\mathbf{x}\cdot\mathbf{x
}$ not a linear function?
Thanks


If an inner product is linear by definition ...

No, an inner product $\langle ,\cdot,\rangle$ of a (real) vector space $V$, is bilinear, not linear. It is linear in each component.
Moreover, the domain of an inner product on $V$ is $V\times V$, while that of a linear map on $V$ is $V$. These two objects have completely different domains.
