2.2 Linear Transformation in Geometry Example. 1 Consider a linear ...
2.2 Linear Transformation in Geometry Example. 1 Consider a linear ... 2.2 Linear Transformation in Geometry Example. 1 Consider a linear ...
Definition. orthogonal projection Let L be a line in R n consisting of all scalar multiples of some unit vector u. For any vector v in R n there is a unique vector w on L such that v − w is perpendicular to L, namely, w = (u·v)u. This vector w is called the orthogonal projection of v onto L: proj L(v) = (u · v)u The transformation proj L from R n to R n is linear. 8
Definition. Let L be a line in R n , the vector 2(proj Lv) − v is called the reflection of v in L: ref L(v) = 2(proj Lv) − v = 2(u · v)u − v where u is a unit vector on L. ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏L ✏✏✏✶ ✏✏✏✏✏✏✏✏✏✏✶ projLv ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✶ ❇▼ ❇ ❇ 2projLv ❇ ❇ ❇ ❇ ★ ❇ ★★★★★★★★❲ v ★ ★★★★★★★★❲ ✲ ✲ u refLv Homework. Exercise 2.2: 1, 9, 13, 17, 27 9
- Page 1 and 2: 2.2 Linear Transformation in Geomet
- Page 3 and 4: Example. 2 Consider a linear transf
- Page 5 and 6: [Shears] Example. 6 Consider the li
- Page 7: Consider a line L in R 2 . For any
Def<strong>in</strong>ition. orthogonal projection Let L be<br />
a l<strong>in</strong>e <strong>in</strong> R n consist<strong>in</strong>g of all scalar multiples<br />
of some unit vector u. For any vector v <strong>in</strong> R n<br />
there is a unique vector w on L such that v − w<br />
is perpendicular to L, namely, w = (u·v)u. This<br />
vector w is called the orthogonal projection<br />
of v onto L:<br />
proj L(v) = (u · v)u<br />
The transformation proj L from R n to R n is l<strong>in</strong>ear.<br />
8
Change language