Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

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.

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

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?
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 $$\vec{a}_2$$."
• 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? – KGhatak May 20 at 17:53