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I have only started linear algebra, I was watching the lectures of Gilbert Strang, he gave us two linear equation
$2x-y=0$
$x-2y=3$ .
He wrote this in the way of coefficient matrices and it became

$\begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix} $ $\begin{bmatrix} x \\ y \end{bmatrix} $=$\begin{bmatrix} 0\\ 3 \end{bmatrix}$.

Now he wrote this as $\begin{bmatrix}2 \\ 1 \end{bmatrix}$$x$ +$\begin{bmatrix} -1\\ -2 \end{bmatrix}$$y$ =$\begin{bmatrix} 0 \\ 3 \end{bmatrix}$.

Next he made the matrix \begin{bmatrix} 2\\ 1 \end{bmatrix} into a vector with components $2$ and $1$. Now I do not get how can you do that aren't these just coefficients matrices which are basically scalars multiplied by components of vectors which are $x$ and $y$. Can someone tell why has it been done like this??

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  • $\begingroup$ I am not too sure as to what do you not understand properly. I am writing a general idea as how to interpret the vectors. There are multiple ways to see how one can get a vector associated with x and y. First, if you just see that a vector times a number is nothing but a new vector with each entry multiplied by that number. Second, if you see the matrix, entries in the first column are multiplied by x and second column with y. Therefore, one can decompose the matrix by writing out two vectors and associating variables x and y with them. $\endgroup$
    – Tushar Pandey
    Jul 14, 2020 at 5:25
  • $\begingroup$ @TusharPandey The coefficient matrix is nothing for than numbers which are multiplied by the x component of the components of vectors, what does the x component multiplied by a matrix as i have given 2 and 1 even mean, there are not vectors, they are just numbers how can he plot them like vectors with component 2 and 1 in the first column $\endgroup$
    – BlackThunder
    Jul 14, 2020 at 5:57
  • $\begingroup$ vector in $R^2$ is just a tuple/pair of points. The matrix is a way of representing the map from $R^2$ to $R^2$. If you have one equation, say $2x + 3y = 4$, you can represent it as $\begin{bmatrix} 2 \end{bmatrix} x +\begin{bmatrix} 3 \end{bmatrix}y = \begin{bmatrix} 4 \end{bmatrix} $. If you have two such equations, you can stack them up. So, the vector is just to represent that there are two equations and this is what the combined form looks like. $\endgroup$
    – Tushar Pandey
    Jul 14, 2020 at 14:05

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$$\begin{bmatrix} x \\ y \end{bmatrix} =x\begin{bmatrix} 1\\ 0 \end{bmatrix} +y\begin{bmatrix} 0\\ 1 \end{bmatrix} $$

So $$ \begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} =x \begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} 1\\ 0 \end{bmatrix} +y\begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix}\begin{bmatrix} 0\\ 1 \end{bmatrix} $$

$$= x \begin{bmatrix} 2 \\ 1 \end{bmatrix} +y\begin{bmatrix} -1 \\ -2 \end{bmatrix} $$

Don't think of $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ as two of the four components in the matrix, think of it as the vector you get when you use the matrix to transform the unit vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$

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  • $\begingroup$ But I do not get how can a unit vector have two components,x only has one component ,how can you plot the matrix 2 and 1, $\endgroup$
    – BlackThunder
    Jul 14, 2020 at 13:15

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