What is the difference between these notations:
\begin{equation*}
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\end{equation*}
And:
\begin{equation*}
x
\begin{pmatrix}
1 \\
0 \\
\end{pmatrix}
+
y
\begin{pmatrix}
0 \\
1 \\
\end{pmatrix}
\end{equation*}
And:
\begin{equation*}
\langle1,0 \rangle x + \langle0,1 \rangle y
\end{equation*}
Are they all the same? Why is vector notation sometimes used and matrix notation sometimes used?
Best Answer
The latter two notations are the same if one takes $\langle a,b \rangle :=\begin{pmatrix} a\\ b \end{pmatrix}$. This notation is sometimes used to have vectors written in-line. In this case, what you have is a linear combination of the two standard vectors:
$$x\langle 1,0 \rangle+y\langle 0,1 \rangle =x\begin{pmatrix} 1\\ 0 \end{pmatrix} + y\begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} x\\ y \end{pmatrix} = \langle x,y \rangle$$ Notice that these are both notations for vectors. The first notation is for a 2-by-2 matrix, which is not a vector unlike the latter two expressions. You can write the latter two expressions using the matrix-vector product, which might be the source of your confusion. In the 2-by-2 case the matrix-vector product is defined $$\begin{bmatrix} a &b \\ c & d \end{bmatrix}\begin{pmatrix} x\\ y \end{pmatrix} =x\begin{pmatrix} a\\ c \end{pmatrix}+y\begin{pmatrix} b\\ d \end{pmatrix}$$ From this it should be trivial to see what 2-by-2 matrix is used to give you your latter two expressions.