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

[@lindamanas6735]

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.

[@ALevelMathsRevision]

Yes the textbook does seem to be lacking on this front.

[@vladid5896]

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.

[@ALevelMathsRevision]

Thank you for the alternative take on things. I'm sure lots of people will find it useful.

[@kazishahjalal6852]

Your method is great but clearly it doesn't explain why m is not equal to -1/3. You can put m=3/4 and -4/3 in equation 2 and still find they are not equal. Thanks for sharing your method though.

[@himanshu7544]

Lifesaver

[@awall007]

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?

[@melodyzhang6969]

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

[@melodyzhang6969]

Nvm I figured it out

[@richardvolkov4867]

thanks

[@ALevelMathsRevision]

Glad to help

[@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.

[@ALevelMathsRevision]

The chain of implications certainly holds and it scores full marks in an exam.

[@clement_jacob]

@@ALevelMathsRevision ok ty for the comment 👍😁. if I instead chose to find the invariant lines normally in that situation would I still score top marks?

[@ALevelMathsRevision]

@@clement_jacob what do you mean by "normally"? Any valid method scores full marks.

[@ALevelMathsRevision]

Also, keep arguing to the death. It's what makes a good mathematician 🙂

[@clement_jacob]

@@ALevelMathsRevision as in like you know how you solved it in the second example, if I attempted and solved the question one in this video just like you solved questions 2 I would get all marks right ? ty for helpful comments btw I appreciate it