# Relation between system of linear equations and Linear transformations

can anyone explain the relation between system of linear equations and Linear transformations
3a + b = -3
a + 2b = 4
Can the following statement be correct ?
Here (-3,4) are linear transformations of solution vector ( i.e; solution that we get after solving of system of linear equations ) ?

I’m not a mentor for that course, but I’ll give your question a try.

A linear transformation would be (for example) if you get a new equation by simply multiplying both sides of an equation by a constant.

So if you take the 2nd equation, and multiply both sides by 3, you’d get 3a + 6b = 12.
Since the first equation has just ‘b’, not ‘6b’, it isn’t a linear transformation of the 2nd equation.

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Now, if you try to solve that system of equations, you’d continue by subtracting equation 2 from equation 1.

This would give you -5b = -15, then divide both sides by -5, and you’re get b = 3.

Now that you know b = 3, you can substitute it into either equation and learn that a = -2.

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Thank you TMosh for replying,
I was asking about Linear transformation and you told me about linear dependency.

Hello sai,
linear equation its a simple form of line equation ,y=mx+c (m=slope,c=intercept).But here y=mx+c is a function in which x values (scalars) are independent and y values are dependent.
But in linear transformation first we are dealing with vectors. That means we transforming vectors with help of linear transforming matrix from one basis to other basis.
you can compare this with linear equation y=transformed vector [-3,4],x=transformation matrix or new basis of transformation[[3,1],[1,2]] , m=ta vector which we want to transfer [[a],[b]]
the ans for your question is , the vector [a,b] will take transfor the [a,b] vector into [-3,4] with the help of transformation matrix
i hope you understood

I think you are right you can think of it as [3, 1] * a + [1, 2] * b = [-3, 4] you are now having a basis of [3, 1] , [1, 2] of the space and you are simply saying well what is the linear combination of those basis that will give me [-3, 4]? and turns out that the solution is undoing the original linear transformation you have done for example if your transformation is rotation then the solution of the system is undoing the rotation which “undoing” have it’s matrix in another word it’s inverse.