Thread: Points transformed by a matrix.

So, my semester is ending in less than one week, and I've been studying for my Linear Algebra exam, but I can't seem to solve this problem.

A, B, C -> 3 points [x, y];
M -> a transformation Matrix.

Demonstrate that if A, B and C are aligned, MA, MB and MC are also aligned.

I.e : a line, defined by a 2*n matrix of points, will stay a line after being transformed by a matrix.


If anyone could help me, I'd appreciate it, I've tried searching for a similar thread, but nothing came up. I hope it's clear enough

Also, feel free to correct my English, the terminology i used might not be correct since it's directly translated from French.

Points A, B, C aligned means the difference vectors are parallel, i.e. A - B = k(B - C)

Now, can you show that MA, MB, MC are aligned by using this? I.e. MA - MB = k'(MB - MC)

Recall that a linear transformation has the property M(cA + c'B) = cMA + c'MB.

Thanks, that's probably what I have to use. But I don't recall seeing this property (M(cA + c'B) = cMA + c'MB) in class, could you explain the meaning of c and c'?

c and c' are just multiplicative constants (i.e. scalars / field elements).

This property is actually at the heart of linear transformations and matrices. The only reason why matrix multiplications works the way it does is because of this property.