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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?

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    $\begingroup$ They are the same. A linear transformation defined by a matrix is equivalent to the span of the column vectors. $\endgroup$ – Ty Jensen Feb 1 '20 at 4:49
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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.

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They are actually all the same! It is only a way of representation! If you take the first one, you'll have a matrix with just numbers that represent linear combinations!

The second one actually shows linear combinations in the form: cw + dv; where w and v are the vectors! And C and D are the scalars, which in your case is x and y.

And the angled bracket you see "<>" represents an inner product, for example, the scalar product or the dot product

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