Finding General Invariant Lines Under Matrix Transformations [Yr1 (Further) Pure Core]

An A Level Further Maths Revision Tutorial on Finding General Invariant Lines Under Matrix Transformations

[lindamanas]

Excellent! This has been omitted from the Edexcel textbook. Whoever wrote that chapter should be ashamed of themselves. Should not have been allowed without further revisions to the text.

[vladid]

Great video, but I think you might have gone the hard way to find the invariant lines of your matrices. Taking your (7 24 | 24 -7) matrix, you can make the problem much more simple by comparing coefficients. At the stage where you have 24x - 7mx - 7c = 7mx + 24m2x + 24mc + c, you can approach it differently: without taking the terms to one side, by grouping the x terms and the c terms as follows: (24 - 7m)x - 7c = (7m + 24m2)x + (24m +1)c ; you can compare coefficients. This gives you 2 equations; (1.) 24 - 7m = 7m +24m2
24m2 + 14m - 24 = 0 (Taking the terms to one side)
m = 3/4 or -4/3 (Done by quadratic formula)
and
(2.) -7c = (24m + 1)c
(24m + 8)c = 0 (Taking the terms to one side)
c = 0 (ignoring 24m + 8 = 0 as m is shown NOT to equal -1/3 in equation (1.))

Hence invariant lines are:
y = 3/4 x
y = -4/3 x
I personally find this easier, but this might not be the case for everyone. I hope this helps.

[awall]

When you plug in x=0 and m=-1/3 to y=mx + c then you get y = c where x = 0 and c is free. Why does this imply only the origin is invariant instead of the y axis being invariant?

[melodyzhang]

What happens if the question asks you to find the invariant line but with a condition of c is not 0?

[clement_jacob]

hmm not sure about this, you said you need to prove y prime = x prime + c, but you used this statement in your proof itself. Isnt this counter intuitve your proofing something by using itself in the actual proof.