In the linear algebra course I am taking, the inner product of 2 vectors $\langle u, v \rangle$ is defined as being a scalar; however, it is also viewed as being a product of 2 matrices as $u^Tv$, as vectors are said to be a specific type of matrix, and the product of 2 matrices is defined as being a matrix. This seems to suggest a scalar is a matrix; yet matrix-scalar multiplication doesn't make sense if the scalar is a matrix.
So my question is how to reconcile how in one case a scalar seems like it is a matrix, and on the other hand seems like it isn't, based on the way these concepts are expounded in the specific course I am taking. Is it the case that a scalar is indeed a matrix, but matrix-matrix multiplication is just defined in a unique way in the case of one of the factors being a scalar, which differs from the standard view?
Best Answer
If $u$ and $v$ are vectors in $\mathbb{R}^n$ then we can view them as $n \times 1$ matrices. Then $u^Tv$ is a $1 \times 1$ matrix, so it looks like $[\alpha]$ for some $\alpha \in \mathbb{R}$. Because of how matrix multiplication and the standard inner product are defined, the number $\alpha$ is exactly $\alpha = \langle u, v \rangle$. So strictly speaking, $u^Tv$ and $\langle u, v \rangle$ are different things, because $u^Tv$ is a $1 \times 1$ matrix, while $\langle u, v \rangle$ is a scalar. However, since the only number in the matrix $u^Tv$ is $\langle u, v \rangle$, it is common to treat them as interchangable.