MATA23H3 Lecture Notes - Lecture 15: Hyperplane, Elementary Matrix, Diagnostic And Statistical Manual Of Mental Disorders
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The non-invertible linear transformations (cid:1846): (cid:2870) (cid:2870) are usually called collapsing linear. They collapse (cid:2870)into a line or a point. Column space is a 1-dim subspace of (cid:2870) Btw, there are other collapses to a line that are not projection. If it has a 2nd non-zero column, it must be a multiple of the first one. Is projected onto (cid:1868) on a line if (cid:4666) (cid:1868) (cid:4667) (cid:1868) . If (cid:1846): (cid:2870) (cid:2870) is a rotation of the plane through angle in the counterclockwise direction, then the standard matrix representation of (cid:1846) is =[(cid:1855)(cid:1867)(cid:1871) (cid:1871)(cid:1861)(cid:1866) (cid:1855)(cid:1867)(cid:1871)] (cid:1871)(cid:1861)(cid:1866) Example: reflection in a line (cid:1846)(cid:4666)(cid:1857) (cid:2869)(cid:4667) (cid:1857) (cid:2869) (cid:1855)(cid:1867)(cid:1871) (cid:1846)(cid:4666)(cid:1857) (cid:2870)(cid:4667) (cid:1846)(cid:4666)(cid:1857) (cid:2869)(cid:4667)=(cid:4666)(cid:1855)(cid:1867)(cid:1871)(cid:4667)(cid:4666)(cid:1857) (cid:2869)(cid:4667)+(cid:4666)(cid:1871)(cid:1861)(cid:1866)(cid:4667)(cid:4666)(cid:1857) (cid:2870)(cid:4667)=[(cid:1855)(cid:1867)(cid:1871),(cid:1871)(cid:1861)(cid:1866)] (cid:1846)(cid:4666)(cid:1857) (cid:2870)(cid:4667)=(cid:4666) (cid:1871)(cid:1861)(cid:1866)(cid:4667)(cid:1857) (cid:2869)+(cid:4666)(cid:1855)(cid:1867)(cid:1871)(cid:4667)(cid:1857) (cid:2870)=[ (cid:1871)(cid:1861)(cid:1866),(cid:1855)(cid:1867)(cid:1871)] =[(cid:1846)]=[(cid:1855)(cid:1867)(cid:1871) (cid:1871)(cid:1861)(cid:1866) (cid:1855)(cid:1867)(cid:1871)] can show that (cid:1846) is a linear transformation (cid:1871)(cid:1861)(cid:1866) We can regard a clockwise direction through an angle as a counterclockwise through an angle of .
