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matrices

can some please explain part a, and b? thanks
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Avatar for mqb2766
mqb2766
21
Original post by FurtherMaths2020
can some please explain part a, and b? thanks


a) Consider the matrix multiplying the column vector (x1,y1). What is the result?
b) The new matrix must do something similar to a) bit introduces a multiplier "m". Any ideas?
Original post by FurtherMaths2020
can some please explain part a, and b? thanks


a) Take any general point (x,y)(x,y), apply the map to it, and argue that it lies on the line y=xy=x.

b) Use the matrix in part (a) but adjust it slightly so that it maps the general point (x,y)(x,y) onto y=mxy=mx.

c) A linear map must satisfy two conditions. What are they? Can you show that one of them fails if any point goes to ax+by=cax+by = c ?
Original post by mqb2766
a) Consider the matrix multiplying the column vector (x1,y1). What is the result?
b) The new matrix must do something similar to a) bit introduces a multiplier "m". Any ideas?


ok, for part a, if I multiply a by (a, b) I get (b, b), in which case, y = x.

as for b, replace (1,1) with (m ,1), and I get ( bm, b)?

whats going on with c? I dont understand what the question is saying, nor asking.
(edited 4 years ago)
Original post by RDKGames


c) A linear map must satisfy two conditions. What are they? Can you show that one of them fails if any point goes to ax+by=cax+by = c ?


thanks for part a and b,

for c: for the line to be linear, the gradient must be constant. no, I don't know how to show that.
Original post by FurtherMaths2020
thanks for part a and b,

for c: for the line to be linear, the gradient must be constant. no, I don't know how to show that.


No that's not it.

Go back through your textbook. There *must* be a definition for a linear transformation / mapping in there that you should know.
Original post by RDKGames
No that's not it.

Go back through your textbook. There *must* be a definition for a linear transformation / mapping in there that you should know.


any linear transformation always map the origin onto itself, and any linear transformation can be represent by a matrix.
Original post by FurtherMaths2020
any linear transformation always map the origin onto itself, and any linear transformation can be represent by a matrix.


I was thinking more along the lines of two conditions ... T is linear if:

T(α+β)=T(α)+T(β)T(\alpha + \beta) = T(\alpha) + T(\beta)
and
T(λα)=λT(α)T(\lambda \alpha) = \lambda T(\alpha)

But fine, you can use the fact that the origin must map back onto itself.

So go ahead and apply the map to the origin and argue that the result is not on the line ax+by=cax + by = c where c≠0c \neq 0.
Reply 8
Avatar for mqb2766
mqb2766
21
Original post by FurtherMaths2020
ok, for part a, if I multiply a by (a, b) I get (b, b), in which case, y = x.

as for b, replace (1,1) with (m ,1), and I get ( bm, b)?

whats going on with c? I dont understand what the question is saying, nor asking.

Is your matrix the right way round for b)?
Original post by mqb2766
Is your matrix the right way round for b)?

here, y = 6, and x = mb?

ok, I see where I've gone wrong. Y = mb, and x = b, in the equation: y = mx

so, ive put the m in the wrong place.
(edited 4 years ago)
Original post by RDKGames
I was thinking more along the lines of two conditions ... T is linear if:

T(α+β)=T(α)+T(β)T(\alpha + \beta) = T(\alpha) + T(\beta)
and
T(λα)=λT(α)T(\lambda \alpha) = \lambda T(\alpha)

But fine, you can use the fact that the origin must map back onto itself.

So go ahead and apply the map to the origin and argue that the result is not on the line ax+by=cax + by = c where c≠0c \neq 0.


what does it mean - a linear matrix always maps origin onto itself?
Original post by FurtherMaths2020
what does it mean - a linear matrix always maps origin onto itself?

A matrix M always satisfies
0 = M0
So it maps the origin to the origin. The matrix just scales (and adds) individual entries.
Original post by mqb2766
A matrix M always satisfies
0 = M0
So it maps the origin to the origin. The matrix just scales (and adds) individual entries.

ah, ok!

so, c must equal c? because otherwise, the transformation that satisfies that equation doesnt pass through the origin and doesnt scale individual entries, but moves them?
Original post by mqb2766
A matrix M always satisfies
0 = M0
So it maps the origin to the origin. The matrix just scales (and adds) individual entries.


so, y = mx is a linear transformation and when x = 0, y = 0. when y = 0, x = 0? therefore, it passes through origin?
Original post by FurtherMaths2020
ah, ok!

so, c must equal c? because otherwise, the transformation that satisfies that equation doesnt pass through the origin and doesnt scale individual entries, but moves them?


It's not too difficult, let's be clear about this ...

A linear map must map (0,0) to (0,0). So if, after the transformation, you have (0,0) you should check whether it is on the line ax+by=cax+by=c. Is it?

Well, subbing the coordinates in you get 0=c0=c.

So for a general c∈Rc \in \mathbb{R} this condition does not hold. Therefore the transformation is generally not linear.
(edited 4 years ago)
Original post by RDKGames
It's not too difficult, let's be clear about this ...

A linear map must map (0,0) to (0,0). So if, after the transformation, you have (0,0) you should check whether it is on the line ax+by=cax+by=c. Is it?

Well, subbing the coordinates in you get 0=c0=c.

So for a general c∈Rc \in \mathbb{R} this condition does not hold. Therefore the transformation is generally not linear.


ok, it must be linear and pass through the origin. so when y = x, and when x =0, y =0.

if it doesnt pass through the origin, then: ax + by = c, when x = 0, y = c, not zero (unless c is 0), therefore, ax + by = c is not a linear transformation unless c = 0 (passes through origin)?
(edited 4 years ago)
Original post by FurtherMaths2020
ok, it must be linear and pass through the origin. so when y = x, and when x =0, y =0.

if it doesnt pass through the origin, then: ax + by = c, when x = 0, y = c, not zero (unless c is 0), therefore, ax + by = c is not a linear transformation unless c = 0 (passes through origin)?


Most of what you've just said isn't very coherent, so sorry I don't even follow your argument.
Original post by RDKGames
Most of what you've just said isn't very coherent, so sorry I don't even follow your argument.


yeah, sorry - I meant: we know when a linear transformation is in the form of y = x, that when y = 0, x = 0, therefore it passes through the origin.

in the form of y = x + c, c must equal 0 (pass through origin), otherwise, when x = 0, y doesn't equal 0, but c.

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