How does the dot product convert a matrix into a scalar? 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.
 A: It is true that you can only multiply a $m \times n$ matrix by a $n \times p$ matrix, i.e., the column size of the left matrix has to match the row size of the right matrix. With this, we can conclude that a product of a $1 \times 1$ matrix by a $n \times p$ matrix makes no sense for $n > 1$.
That being said, the space of matrices is a vector space, so it has the multiplication between scalars and matrices. So it makes sense to multiply a scalar by a matrix. Once you realize the space of scalars and $1 \times 1$ matrices can be identified, the confusion goes away.
A: There are two ways to look at dot products:


*

*The scalar product of two vectors $(v_1,...,v_n)$ and $(w_1,...,w_n)$ can be simply defined as the sum $v_1w_1+\cdots+v_nw_n$, and so its a scalar by definition, or as you learned it

*as a special kind of matrix multiplication.


In the latter case, the result is indeed a $(1\times 1)$-matrix, and as the comments already stated, its just a convention to call this a scalar, as "the only difference are the brackets". Of course, when you view your result as a matrix, you can only multiply it by $(1\times m)$-matricies. But here is the clue: the space of $(1\times 1)$-matricies is (for all practical purpose) equivalent to the space of scalars. This means, there is a bijective linear map from $K$ to $K^{1\times1}$, even an algebra isomorphism. So, to use the result in more general fashion, we allow us to interpret it as a scalar. Further, you often use the result of a dot product only to multiply it with vectors. And vectors are not so far from the $(1\times m)$-matricies where you are allowed to use your $(1\times1)$-matricies anyway.
A: Sometimes, we just say that a $1\times 1$ matrix is the same as a scalar.  Afterall, when it comes to addition and multiplication of $1\times 1$ matrices vs addition and multiplication of scalars, the only difference between something like $\begin{bmatrix}3\end{bmatrix}$ and $3$ is some brackets.  Consider $$(3+5)\cdot 4 = 32 \\ (\begin{bmatrix} 3\end{bmatrix} + \begin{bmatrix} 5\end{bmatrix})\begin{bmatrix} 4\end{bmatrix} = \begin{bmatrix} 32\end{bmatrix}$$ The algebra works out exactly the same.  So sometimes it's not ridiculous to think of $1\times 1$ matrices as just another way of writing scalars.
But if you do want to distinguish the two, then just think of the formula $a\cdot b = a^Tb$ as a way of finding out which scalar you get from the dot product of $a$ and $b$ and not literally the dot product value itself (which should be scalar).  That is, we calculate the dot product of $\begin{bmatrix} 1 \\ 2\end{bmatrix}$ and $\begin{bmatrix} 3 \\ 4\end{bmatrix}$ by using the formula $$\begin{bmatrix} 1 \\ 2\end{bmatrix}^T\begin{bmatrix} 3 \\ 4\end{bmatrix} = \begin{bmatrix} 1 & 2\end{bmatrix}\begin{bmatrix} 3 \\ 4\end{bmatrix} = \begin{bmatrix} 11\end{bmatrix}$$ and then say that this tells us that the dot product is really $11$.  So the formula $a^Tb$ is just an algorithm we use to find the correct scalar.
You can view it either way.  It doesn't really make a difference.
A: You are not wrong and it's always good to examine statements very carefully. We usually do not distinguish 1$\times$1 matrices from scalars, and in some sense you can think of scalar multiplication as a special rule for when one of the matrices is 1$\times$1. But the truth is that it is a convenient abuse of notation.
Here's one other thing to think about. A $1 \times 1$ real matrix is supposed to represent a linear mapping from a one-dimensional real vector space into a one-dimensional real vector space--essentially just $\mathbb R$ into $\mathbb R$. The only such mappings are those that take $x \mapsto ax$ for some fixed real number $a$. But this mapping is entirely determined by that real number $a$, so they are essentially equivalent. 
A: A dot product is not really a scalar, but it behaves just like one.   In math we call that an ISOMORPHISM.  For every dot product result, there is a corresponding real number that you get by simply removing the brackets.   All of the operations you do with the 1x1 matrix correspond to the same operations done with a real number.  Multipliction of a matrix by a 1x1 matrix is defined as multiplcation by the corresponding scalar.   
Recommend you look up "isomorphism" in a math dictionary.   To do this, you may need to brush up on your German language skills.
