1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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 $\vec{a}_2$."

$\endgroup$
  • $\begingroup$ 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 May 20 at 17:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.