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While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals that $\vec {a_1}$ and $\vec {a_2}$, the columns of A, are vectors.

For the matrix equation, I interpret $\vec b$ as a linear transformation of $\vec x$. So I realized that they don't need to be in the same dimensional system.

However, I'm not sure how to relate $\vec {a_1}$ or $\vec {a_2}$ with $\vec b$? I'm not even sure how to make sense out of the expression of the linear combination, which uses scalar components of the vector $\vec x$?

In fact, I took below concrete examples to visualize but not with much luck. Any input is much appreciated! Any input/example form Physics/Chemistry/etc. will be very useful.

$$A=\begin{bmatrix} 1 & -3\\ 3 & 5 \\ -1 & 7 \end{bmatrix} x=\begin{bmatrix} {x_1}\\ {x_2} \end{bmatrix} b=\begin{bmatrix} 3\\ 2\\ -5 \end{bmatrix}$$


Added on 21-May-2019:

Specifically, my concern is more around coordinate systems to represent the vectors e.g. to add two vectors I typically imagine a common coordinate system to represent both the vectors. Here - (i) For $A \vec x=\vec b$ , I feel we need two different coordinate systems to represent $\vec x$ and $\vec b$ separately - is it so? (ii)For linear combination form, do we need a 3rd coordinate system to represent $\vec {a_1}$ and $\vec {a_2}$. If so, then how do these coordinate systems relate to each other?

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A linear equation of the form $$ \underbrace{\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nn} \end{bmatrix}}_{A} \underbrace{\begin{bmatrix} x_{1}\\ \vdots\\ x_{n} \end{bmatrix}}_{\vec x}= \underbrace{\begin{bmatrix} b_1\\ \vdots\\ b_n \end{bmatrix}}_{\vec b}$$ has also the following interpretation: let $A$ have columns $\vec{a}_1,\ldots, \vec{a}_n$. Can we find $n$ scalars $x_1,\ldots, x_n$ so that $x_1\vec{a}_1+\cdots+x_n\vec{a}_n=\vec b$? That is, if we fix some vector $\vec b\in \mathbb{R}^n$, in what ways can we take weighted sums of the $\vec{a}_i$ to make $\vec b$? So, the linear combination should be interpreted as rescaling the $\vec{a}_i$ and adding them.

For instance, a linear combination of the form $\frac{1}{2}\vec{a}_1+\vec{a}_2$ means "take the vector with the same direction as $\vec{a}_1$ but half of the magnitude and add it to $\vec{a}_2$."

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  • $\begingroup$ My concern is more around coordinate systems to represent the vectors e.g. to add two vectors I typically imagine a common coordinate system to represent both the vectors. Here - (i) For A$\vec x$= $\vec b$, I feel we need two different coordinate systems to represent $\vec x$ and $\vec b$ separately - is it so? (ii)For linear combination form, do we need a 3rd coordinate system to represent ${a_1}$ and ${a_2}$. If so, then how do these coordinate systems relate to each other? $\endgroup$
    – KGhatak
    Commented May 20, 2019 at 17:53

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