Vector equation vs Matrix equation

[Instinctlol]

I am a little bit confused about these 2.
A vector equation goes like this: x₁v₁+x₂v₂+...+x_nv_n=b
and Matrix equation goes like this:
Ax= [v₁ v₂ .. v_n][x₁
x₂
.
.
.
x_n]

v is a vector in the vector equation, but in the matrix equation x becomes the vector and v is just the matrix.

My question is, why does it get switched like this?

 

[Mark44]

I am a little bit confused about these 2.
A vector equation goes like this: x₁v₁+x₂v₂+...+x_nv_n=b
and Matrix equation goes like this:
Ax= [v₁ v₂ .. v_n][x₁
x₂
.
.
.
x_n]

v is a vector in the vector equation, but in the matrix equation x becomes the vector and v is just the matrix.

My question is, why does it get switched like this?

There are different kinds of multiplication going on. In your first equation, each term is a scalar times a vector (I'm assuming that the x_i's are scalars).

Your second equation is the product of what appears to be an n X n matrix and a vector with n components. If you switched the order in the 2nd equation to xA, the multiplication would not be defined.

 

[Deveno]

in your "vector equation" the v's are vectors, and the x's are field elements (scalars).

one way of thinking of a matrix is "a vector of vectors" (it's a 2-array, not a 1-array). so in your "matrix equation" each v is a column (which is why matrix entries need TWO subscripts, one for the row, and one for the column). so what you really have is:

$$\begin{bmatrix}v_{11}&v_{12}&\dots&v_{1n}\\v_{21}&v_{22}&\dots&v_{2n}\\ \vdots&\vdots&\ddots&\vdots\\v_{m1}&v_{m2}&\dots&v_{mn}\end{bmatrix} \begin{bmatrix}x_1\\x_2\\ \vdots\\x_n\end{bmatrix} = \begin{bmatrix}v_{11}x_1 + v_{12}x_2 + \dots + v_{1n}x_n\\v_{21}x_1 + v_{22}x_2 + \dots + v_{2n}x_n\\ \vdots\\v_{m1}x_1 + v_{m2}x_2 + \dots + v_{mn}x_n\end{bmatrix} = \begin{bmatrix}b_1\\b_2\\ \vdots\\b_n\end{bmatrix}$$

writing Ax = b is just an abbreviation for that god-awful mess above.