Core Pure 1 Chapter 7: Linear Transformations

22 flashcards · Mathematics · by TheoTyson

1. What is an image?

   A point with a transformation applied.

2. What are linear transformations?

   Map the origin onto itself. Represented by a matrix.

3. What can the linear transformation TTT ::: xy\begin{matrix} x \\ y \end{matrix}xy​ ↦\mapsto↦ axbycxdy\begin{matrix} ax & by \\ cx & dy \end{matrix}axcx​bydy​ be represented by?

   MMM === abcd\begin{matrix} a & b \\ c & d \end{matrix}ac​bd​

4. What are invariant points?

   Points which are mapped onto themselves under the given transformation.

5. What are invariant lines?

   Lines which are mapped onto themselves under the given transformation.

6. What matrix represents a reflection in yyy -axis?

   −1001\begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix}−10​01​ Points on yyy -axis are invariant. Lines x=0x=0x=0 and y=ky=ky=k are invariant.

7. What matrix represents a reflection in xxx -axis?

   100−1\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}10​0−1​ Points on xxx -axis are invariant. Lines y=0y=0y=0 and x=kx=kx=k are invariant.

8. What matrix represents a reflection in the line y=xy=xy=x ?

   0110\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}01​10​ Points on y=xy=xy=x are invariant. Lines y=xy=xy=x and y=−x+ky=-x+ky=−x+k are invariant.

9. What matrix represents a reflection in the line y=−xy=-xy=−x ?

   0−1−10\begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix}0−1​−10​ Points on y=−xy=-xy=−x are invariant. Lines y=−xy=-xy=−x and y=x+ky=x+ky=x+k are invariant.

10. What matrix represents a rotation through θ\thetaθ anticlockwise about the origin?

    cosθ−sinθsinθcosθ\begin{matrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{matrix}cosθsinθ​−sinθcosθ​ The origin is invariant. When θ=180°\theta = 180 \degreeθ=180° any line passing through the origin is invariant.

11. What matrix represents an enlargement with a scale factor of aaa ?

    a00a\begin{matrix} a & 0 \\ 0 & a \end{matrix}a0​0a​ The origin is invariant. The xxx -axis and yyy -axis are invariant lines.

12. What matrix represents a stretch of scale factor aaa parallel to the xxx -axis?

    a001\begin{matrix} a & 0 \\ 0 & 1 \end{matrix}a0​01​ Points on yyy -axis are invariant. Lines parallel to xxx -axis are invariant and both axes.

13. What matrix represents a stretch of scale factor bbb parallel to the yyy -axis?

    100b\begin{matrix} 1 & 0 \\ 0 & b \end{matrix}10​0b​ Points on xxx -axis are invariant. Lines parallel to yyy -axis are invariant and both axes.

14. What is the scale factor for change in area of transformation MMM ?

    Det⁡{\operatorname {Det}}Det MMM

15. What does the matrix PQPQPQ represent?

    Transformation QQQ followed by transformation PPP .

16. What matrix represents a reflection in the plane x=0x=0x=0 ?

    −100010001\begin{matrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}−100​010​001​

17. What matrix represents a reflection in the plane y=0y=0y=0 ?

    1000−10001\begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}100​0−10​001​

18. What matrix represents a reflection in the plane z=0z=0z=0 ?

    10001000−1\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix}100​010​00−1​

19. What matrix represents a rotation θ\thetaθ about the xxx -axis?

    1000cosθ−sinθ0sinθcosθ\begin{matrix} 1 & 0 & 0 \\ 0 & cos \theta & - sin \theta \\ 0 & sin \theta & cos \theta \end{matrix}100​0cosθsinθ​0−sinθcosθ​

20. What matrix represents a rotation θ\thetaθ about the yyy -axis?

    cosθ0sinθ010−sinθ0cosθ\begin{matrix} cos \theta & 0 & sin \theta \\ 0 & 1 & 0 \\ - sin \theta & 0 & cos \theta \end{matrix}cosθ0−sinθ​010​sinθ0cosθ​

21. What matrix represents a rotation θ\thetaθ about the zzz -axis?

    cosθ−sinθ0sinθcosθ0001\begin{matrix} cos \theta & - sin \theta & 0 \\ sin \theta & cos \theta & 0 \\ 0 & 0 & 1 \end{matrix}cosθsinθ0​−sinθcosθ0​001​

22. What is the transformation described by matrix A−1A^{-1}A−1 ?

    Reverse effect described by transformation AAA .