CH1 Introduction to vectors
The heart of linear algebra is two kinds of operation of vectors: multiply by a const; add two vectors. These two can be combined to give the linear combination cv+dw.
The whole book is going from 2 or 3 dimensions to n dimensions.
1.1 Vectors and linear combinations
Coloumn vector v = [ v1 ; v2 ]; v1:first component; v2: second component.
Vector addition: v = [ v1 ; v2 ], w = [ w1 ; w2 ], v + w = [ v1 + w1; v2 + w2 ].
Scalar Multiplication: 2v = [ 2v1; 2v2 ]
Linear combinations:
DEFINITION: The sum of cv + dw is the linear combination of v and w.
0v + 0w = zero vector
w +v gives the same vector as v + w
Vectors in three dimensions:
There is a perfect match between the column vector and the arrow from the origin and the point where the arrow ends.
From now on v = [ 1 ; 2 ; 3 ] is also written as v = ( 1 , 2 , 3)
v = (1, 2, 3) is not a row vector. vector [1 2 3] is the transpose of v= (1, 2, 3)
v, w , v + w, v - w all lie in the same plane.
The Important Questions
1. The picture of all the combinations of cu -> fill a line.
2. The picture of all the combinations of cu + dv -> fill a plane.
3. The picture of all the combinaitons of cu + dv + ew -> fill three-dimensional space. (R^3)
Worked Examples:
1.1A The linear combinaitons of v = (1, 1, 0) and w = (0, 1, 1) fill a plane. Decribe the plane
1) cv + dw = (c, c, 0) + (0, d, d) = (c, c + d, d); 2) Through (0, 0, 0) because n = (1, -1, 1) is perpendicular to the plane.
1.1B For v = (1, 0) and w = (0, 1), describe all points cv (1) whole numbers c; (2) nonnegative c >= 0. Then add all vectors dw and describe cv + dw.
1) Equally spaced points on x-axis. 1') infinitive paralell lines.
2) fill a half-line, which is the positive x-axis. 2') a half plane.
problem set 1:
（a) line (b) plane (c) all of R^3