How can linear equations be solved using matrices?

[HMPARTICLE]

Right, i don't believe this is a homework question. The only reason I am stating this is because PF are stringent with their rules.

I'm quite confused and I'm not sure how to explicitly state my problem.

The vertices of a triangle are (a,b) (c,d) and (e,f).
This can be arranged into a single matrix.

T = a c e
b d f

with brackets either side i know.

HOWEVER when solving simultaneous equations

2x-3y=5 and -5x+6y=-8

the matrix product

2 -3 X x = 5
-5 6 y 8


is used, so the x values take the first column in this form as oppose to the x values taking the first row in the triangle vertices matrix.


I HAVE TO STATE AGAIN THAT THIS IS NOT A HOMEWORK QUESTION, IT IS JUST A MATTER OF CURIOSITY!

 

[HallsofIvy]

Right, i don't believe this is a homework question. The only reason I am stating this is because PF are stringent with their rules.

I'm quite confused and I'm not sure how to explicitly state my problem.

The vertices of a triangle are (a,b) (c,d) and (e,f).
This can be arranged into a single matrix.

T = a c e
b d f

with brackets either side i know.

Wel, they can be arranged as the arrays
\begin{bmatrix}a & c & e \\ b & d & f \end{bmatrix}
or as
\begin{bmatrix}a & b \\ c & d \\ e & f\end{bmatrix}
But to what purpose? What do you want to do with that array?
(The difference between an "array" and a "matrix" is that a matrix has operations of addition and multiplication defined while an array is simply a way of exhibiting connected numbers.)

HOWEVER when solving simultaneous equations

2x-3y=5 and -5x+6y=-8

the matrix product

2 -3 X x = 5
-5 6 y 8

Yes, using the standard definition of matrix multiplication
\begin{bmatrix}2 & -3 \\ -5 & 6 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2x- 3y \\ -5x+ 6y\end{bmatrix}= \begin{bmatrix}5 \\ 8 \end{bmatrix}

is used, so the x values take the first column in this form as oppose to the x values taking the first row in the triangle vertices matrix.


I HAVE TO STATE AGAIN THAT THIS IS NOT A HOMEWORK QUESTION, IT IS JUST A MATTER OF CURIOSITY!

Curious about what? There is NO question here. What are you trying to ask?

 

[Stephen Tashi]

If you have a situation with data in (x,y) coordinates, there is no general rule about whether the x data must go along a column of a matrix or a row of the matrix.

In solving equations using matrices it's traditional to have a column vector of unknowns multiplied on the left by a matrix, when you do that, you don't have any choice about how you put the coefficients in the matrix since you must recreate the equations when the multiplication is performed. If you went against tradition, you could express the equations as a row vector of unknowns multiplied on the right by a matrix of coefficents. Then you'd have to write the coeffients in the matrix in a different order.

I don't know the context for your example using vertices of a triangle.. Where you reading some math where they must be written into a matrix that way?

 

[HMPARTICLE]

Thanks. So, when vertices of a shape are defined in that way it's called an array. I see now.

The simultaneous equations part is also clear, the l.h.s of the equation is a product of a transformation matrix and a vector.

Thanks for the detailed response!

 

[HMPARTICLE]

To steph it's from a further pure textbook.

 

[HallsofIvy]

By the way, you are mistaken about "the x values take the first column in this form". It was the coefficients of x, not the "x values" that are in the first column.

 

[HMPARTICLE]

Yes thanks for pointing that out! As it's a transformation matrix multiplied by the vector. I'm an undergrad due to start in sept! Just brushing up on my matrices! Thanks guys