• May 22nd 2006, 12:56 PM
Let f be the linear transformation represented by by the matrix

M = 0 2
1 -1

(a) State what effect f has on areas, and whether f changes orientaton.

(b) Find the matrix that represents the inverse of f.

(c) (i) Use the matrix that you found in part b to find the image f (ξ ) of the unit circle ξ under f, in the form

ax^2 +bxy +cy^2 = d

where a, b, c and d are integers.

What is the area enclosed by f ( ξ )
• May 27th 2006, 03:20 AM
CaptainBlack
Quote:

Let f be the linear transformation represented by by the matrix

M = 0 2
1 -1

(a) State what effect f has on areas, and whether f changes orientaton.

Assuming the matrix representation acts on column vectors, the images of
the unit vectors $\displaystyle {1 \brack 0}$ and $\displaystyle {0 \brack 1}$ are $\displaystyle {0 \brack 1}$ and $\displaystyle {2 \brack -1}$ respectively.

So the unit square is transformed into a parallelogram with vertices:

$\displaystyle {0 \brack 1}$,$\displaystyle {0 \brack 0}$,$\displaystyle {2 \brack -1}$, and $\displaystyle {2 \brack 0}$, which has area $\displaystyle 2$.

So the image of a figure under $\displaystyle f$ has twice the area of the figure (as $\displaystyle f$ is linear).

RonL
• May 27th 2006, 05:56 AM
CaptainBlack
Quote:

Let f be the linear transformation represented by by the matrix

M = 0 2
1 -1

(b) Find the matrix that represents the inverse of f.

By Cramer's rule if:

$\displaystyle A=\left[ \begin{array}{cc} a&b\\c&d \end{array} \right]$

then:

$\displaystyle A^{-1}=\left[ \begin{array}{cc} a&b\\c&d \end{array} \right]^{-1}= \left \left[ \begin{array}{cc} d&-b\\-c&a \end{array} \right] \right/ \det(A)=$$\displaystyle \left \left[ \begin{array}{cc} -1&-2\\-1&0 \end{array} \right] \right/ (-2)$

RonL
• May 27th 2006, 06:28 AM
CaptainBlack
Quote:

Let f be the linear transformation represented by by the matrix

M = 0 2
1 -1

(c) (i) Use the matrix that you found in part b to find the image f (ξ ) of the unit circle ξ under f, in the form

ax^2 +bxy +cy^2 = d

where a, b, c and d are integers.

What is the area enclosed by f ( ξ )

Let $\displaystyle {x \brack y}$ be a point on $\displaystyle f(\xi)$, then:

$\displaystyle \left \left[ \begin{array}{cc}-1&-2\\-1&0\end{array} \right]{x \brack y}\right/ (-2)={x/2+y \brack x/2}$,

is a point on the unit circle, so:

$\displaystyle \left(\frac{x}{2}+y\right)^2+\left(\frac{x}{2} \right)^2=1$,

or on rearrangement:

$\displaystyle x^2+2xy+2y=2$.

The area enclosed by the unit circle is $\displaystyle \pi$ so the area
enclosed by $\displaystyle f(\xi)$ is $\displaystyle 2 \pi$.

RonL
• May 27th 2006, 08:07 AM
rgep
In general a linear transformation (of a real Euclidean space) multiplies areas by the determinant. If the determinant is negative, that means that orientation is reversed.
• May 27th 2006, 08:09 AM
CaptainBlack
Quote:

Originally Posted by rgep
In general a linear transformation (of a real Euclidean space) multiplies areas by the determinant. If the determinant is negative, that means that orientation is reversed.