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# P6. Matrices and transforms. watch

1. The linear transformation T from R^3 to R^3 maps the points (1,0) and (1,1) to the points (-2,1) and (-1,1) respectively

Find the 2x2 matrix M which represents T.
Find M and describe T in words
Find the cartesian of the image under T of the line y = x - 1
Firstly, what's R^3?

Now the rest is alright, apart from the last part.

T ends up as

1, -2
0, 1

Now that seems to be a shear parallel to the X axis.

So for the last part

(x,x-1) maps onto (2-x,x-1).

How does I find the cartesian equation of the image of those mappings?
2. R^3, 3 dimensional space.
3. (Original post by JamesF)
R^3, 3 dimensional space.
Thought it might be, but surely this context we're talking R^2?
4. T(a,a-1)=(2-a,a-1)

Doesn't the image thus have cartesian equ^n y=1-x ?
5. (Original post by DanielW3)
T(a,a-1)=(2-a,a-1)

Doesn't the image thus have cartesian equ^n y=1-x ?

Yes. But why? It's not obvious to me.
6. (a,a-1) maps onto (2-a,a-1) so x = 2-a, y = a-1
a = 2-x, a = 1+y
2 - x = 1 + y
y = 1 - x
7. (Original post by Bezza)
(a,a-1) maps onto (2-a,a-1) so x = 2-a, y = a-1
a = 2-x, a = 1+y
2 - x = 1 + y
y = 1 - x
That's what I would've done
8. (Original post by fishpaste)
Firstly, what's R^3?

Now the rest is alright, apart from the last part.

T ends up as

1, -2
0, 1

Now that seems to be a shear parallel to the X axis.

So for the last part

(x,x-1) maps onto (2-x,x-1).

How does I find the cartesian equation of the image of those mappings?
That R^3 maps on to R^3 just means that a dimension isn't crushed in the transformation (ie the determinant of the matrix is not zero).

The linear transformation T from R^3 to R^3 maps the points (1,0) and (1,1) to the points (-2,1) and (-1,1) respectively

Find the 2x2 matrix M which represents T.
Find M and describe T in words
Find the cartesian of the image under T of the line y = x - 1

The linear transformation T from R^3 to R^3 maps the points (1,0) and (1,1) to the points (-2,1) and (-1,1) respectively

Find the 2x2 matrix M which represents T.
Find M and describe T in words
Find the cartesian of the image under T of the line y = x - 1

Once you've found the matrix, you can always use matix algebra and a suitable parametric equation of the line (eg x=a, y=a-1) and then find what it maps to. You'll then get the parametric equation of the line, from which you can find the cartesian equation easily. (x=-a-1 and y=a, so a=-1-x and y=-1-x).

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Updated: June 9, 2004
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