P and Q are two linear transformations in the X-Y plane defined by P: (x, y) → (-3x + 6y, 4x + y) and Q: (x, y) → (2x-3y, -4x - 6y). (a) Write down the matrices of P and Q. (b) What is the image of (-2,-3) under the transformation Q? (c) Obtain a single transformation representing the transformation Q followed by P. (d) Find the image of (1,4) when transformed by Q followed by P. (e) Find the image P\(^1\) of the point (-√2,2√2) under an anticlockwise rotation of 225° about the origin.

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P and Q are two linear transformations in the X-Y plane defined by
P: (x, y) → (-3x + 6y, 4x + y) and
Q: (x, y) → (2x-3y, -4x - 6y).
(a) Write down the matrices of P and Q. (b) What is the image of (-2,-3) under the transformation Q?
(c) Obtain a single transformation representing the transformation Q followed by P.
(d) Find the image of (1,4) when transformed by Q followed by P.
(e) Find the image P

^{1}

of the point (-√2,2√2) under an anticlockwise rotation of 225° about the origin.

Explanation

P: (x, y) → (2x+3y, 3x - y)

(\begin{matrix} x \\ y \end{matrix})

→

(\begin{matrix} - 3 x & 6 y \\ 4 x & + y \end{matrix})

(\begin{matrix} - 3 & 6 \\ 4 & + 1 \end{matrix})

(\begin{matrix} x \\ y \end{matrix})

P =

(\begin{matrix} - 3 & 6 \\ 4 & + 1 \end{matrix})

Q: (x, y) → (2x-3y, -4 - 6y)

(\begin{matrix} x \\ y \end{matrix})

→

(\begin{matrix} 2 x & - 3 y \\ - 4 x & - 6 y \end{matrix})

(\begin{matrix} 2 & - 3 \\ - 4 & - 6 \end{matrix})

(\begin{matrix} x \\ y \end{matrix})

Q =

(\begin{matrix} 2 & - 3 \\ - 4 & - 6 \end{matrix})

(b)

(\begin{matrix} 2 & - 3 \\ - 4 & - 6 \end{matrix})

(\begin{matrix} - 2 \\ - 3 \end{matrix})

(\begin{matrix} 2 [- 2] & - 3 [- 3] \\ - 4 [- 2] & - 6 [- 3] \end{matrix})

(\begin{matrix} - 4 & + 9 \\ 8 & 18 \end{matrix})

(\begin{matrix} 5 \\ 26 \end{matrix})

: The image of (-2,-3) is (5,26)
(c) PQ =

(\begin{matrix} - 3 & 6 \\ 4 & 1 \end{matrix})

(\begin{matrix} 2 & - 3 \\ - 4 & - 6 \end{matrix})

(\begin{matrix} - 3 [2] + 6 [- 4] & - 3 [- 3] + 6 [- 6] \\ 4 [2] + 1 [- 4] & 4 [- 3] + 1 [- 6] \end{matrix})

(\begin{matrix} - 6 - 24 & 9 - 36 \\ 8 - 4 & - 12 - 6 \end{matrix})

(\begin{matrix} - 30 & - 27 \\ 4 & - 18 \end{matrix})

= PQ: (x,y) → (-30x - 27y, 4x - 18y)
(d)

(\begin{matrix} - 30 & - 27 \\ 4 & - 18 \end{matrix})

(\begin{matrix} 1 \\ 4 \end{matrix})

(\begin{matrix} - 30 [1] & + [- 27] [4] \\ 4 [1] & + [- 18] [4] \end{matrix})

(\begin{matrix} - 30 & - 108 \\ 4 & - 72 \end{matrix})

(\begin{matrix} - 138 \\ - 68 \end{matrix})

: Image of the point (1,4) is (-138,-68)
(e) T =

(\begin{matrix} c o s 225 ° & - s i n 225 \\ s i n 225 ° & c o s 225 ° \end{matrix})

(\begin{matrix} - \sqrt 2 / 2 & \sqrt 2 / 2 \\ - \sqrt 2 / 2 & - \sqrt 2 / 2 \end{matrix})

(\begin{matrix} - \sqrt 2 \\ 2 \sqrt 2 \end{matrix})

(\begin{matrix} 1 & + 2 \\ 1 & - 2 \end{matrix})

(\begin{matrix} 3 \\ - 1 \end{matrix})

: The image of P

^{1}

is (3,-1)

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