I am learning linear algebra, and I am a bit confused by the dot product and how the answer to the process turns out to be a scalar rather than a matrix.
For $2$ vectors with $2$ components, I learned that dot product is equivalent to a $1 \times 2$ row vector left multiplied by a $2 \times 1$ column vector. The result of such a multiplication should result in a $ 1 \times 1 $ vector. But I am learning that the dot product somehow transforms the result into a scalar, rather than a $ 1 \times 1 $ vector.
Maybe I am missing something but the difference between a $ 1 \times 1 $ vector and a scalar seems important, because you can multiply a scalar by a matrix of any size, but you can only left-multiply a $ 1 \times 1 $ vector by another matrix with $1$ row.
Thanks for any help in understanding this.