1. Chapter 4 Euclidean n-Space Linear Transformations from to Properties of Linear Transformations to Linear Transformations and Polynomials

2. Definition If n is a positive integer, then an ordered n-tuple is a sequence of n real numbers (a 1,a 2,…,a n ). The set of all ordered n-tuple is called n-space and is denoted by. Note that an ordered n-tuple (a 1,a 2,…,a n ) can be viewed either as a “generalized point” or as a “generalized vector”

3. Definition Two vectors u = (u 1,u 2,…,u n ) and v = (v 1,v 2,…, v n ) in are called equal if u 1 = v 1,u 2 = v 2, …, u n = v n The sum u + v is defined by u + v = (u 1 +v 1, u 1 +v 1, …, u n +v n ) and if k is any scalar, the scalar multiple ku is defined by ku = (ku 1,ku 2,…,ku n ) Remarks The operations of addition and scalar multiplication in this definition are called the standard operations on.

4. If u = (u 1,u 2,…,u n ) is any vector in, then the negative (or additive inverse) of u is denoted by -u and is defined by -u = (-u 1,-u 2,…,-u n ). The zero vector in is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0). The difference of vectors in is defined by v – u = v + (-u) = (v 1 – u 1,v 2 – u 2,…,v n – u n )

5. Theorem 4.1.1 (Properties of Vector in ) If u = (u 1,u 2,…,u n ), v = (v 1,v 2,…, v n ), and w = (w 1,w 2,…,w n ) are vectors in and k and m are scalars, then: a) u + v = v + u b) u + (v + w) = (u + v) + w c) u + 0 = 0 + u = u d) u + (-u) = 0; that is, u – u = 0 e) k(mu) = (km)u f) k(u + v) = ku + kv g) (k+m)u = ku+mu h) 1u = u

6. Definition If u = (u 1,u 2,…,u n ), v = (v 1,v 2,…, v n ) are vectors in, then the Euclidean inner product u · v is defined by u · v = u 1 v 1 + u 2 v 2 +… + u n v n Euclidean Inner Product Example The Euclidean inner product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in R4 is u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

7. It is common to refer to, with the operations of addition, scalar multiplication, and the Euclidean inner product, as Euclidean n-space. Theorem 4.1.2 If u, v and w are vectors in and k is any scalar, then a) u · v = v · u b) (u + v) · w = u · w + v · w c) (k u) · v = k(u · v) d) v · v ≥ 0; Further, v · v = 0 if and only if v = 0 Example (3u + 2v) · (4u + v) = (3u) · (4u + v) + (2v) · (4u + v ) = (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v =12(u · u) + 11(u · v) + 2(v · v)

8. Norm and Distance in Euclidean n-Space We define the Euclidean norm (or Euclidean length) of a vector u = (u 1,u 2,…,u n ) in by Similarly, the Euclidean distance between the points u = (u1,u2,…,un) and v = (v 1, v 2,…,v n ) in defined by

9. Example If u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space

10. Theorem 4.1.3 (Cauchy-Schwarz Inequality in ) If u = (u 1,u 2,…,u n ) and v = (v 1, v 2,…,v n ) are vectors in, then |u · v| ≤ || u || || v || Theorem 4.1.4 (Properties of Length in ) If u and v are vectors in and k is any scalar, then a) || u || ≥ 0 b) || u || = 0 if and only if u = 0 c) || ku || = | k ||| u || d) || u + v || ≤ || u || + || v || (Triangle inequality)

11. a) d(u, v) ≥ 0 b) d(u, v) = 0 if and only if u = v c) d(u, v) = d(v, u) d) d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality) Theorem 4.1.5 (Properties of Distance in Rn) If u, v, and w are vectors in and k is any scalar, then Theorem 4.1.6 If u, v, and w are vectors in with the Euclidean inner product, then

12. Definition Two vectors u and v in are called orthogonal if u · v = 0 Orthogonality Example In the Euclidean space the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1) are orthogonal, since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0 Theorem 4.1.7 (Pythagorean Theorem in ) If u and v are orthogonal vectors in whith the Euclidean inner product, then

13. Alternative Notations for Vectors in It is often useful to write a vector u = (u 1,u 2,…,u n ) in in matrix notation as a Row matrix or a column matrix:

14. If we use column matrix notation for the vectors then Matrix Formulae for the Dot Product

15. A Dot Product View of Matrix Multiplication Let A = [a ij ] be an m×r matrix and B =[b ij ] be an r×n matrix, if the row vectors of A are r 1, r 2, …, r m and the column vectors of B are c 1, c 2, …, c n, then the matrix product AB can be expressed as

16. In particular, a linear system Ax=b an be expressed in dot product form as Where r 1, r 2,…, r m are the row vectors of A, and b 1, b 2, …, b m are the entries of b Example: