lecture_16- Vector Spaces

lecture_16- Vector Spaces - MA 265 LECTURE NOTES:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 13 Real Vector Spaces Vectors in n-Space. Recall that R n is the collection of all real n-vectors: x = x 1 x 2 . . . x n where the entries x i are real numbers. We call this collection (real) n-space . The elements x R n will be called vectors. (If the x i are complex numbers, then the collection of all complex n-vectors is denoted by C n .) Vectors in R 1 . Consider n = 1. Elements x R 1 will be called scalars for reasons to be discussed later in the lecture. Note that we do not use any special font such as x to denote this. We usually draw real numbers on the real number line: x d15 d111 d47 | 1 | | 1 | 2 R 1 Recall that positive is to the right and negative is to the left. Vectors in R 2 . Now consider n = 2. The vectors of interest will be in the form x = bracketleftbigg x y bracketrightbigg where x and y are real numbers. We can draw this as a in the Cartesian plane by identifying x with the point P = ( x,y ). These will be called the coordinates of x . Conversely, say that we have two points P = ( x,y ) and Q = ( x ,y ). We can consider the line segment between P and Q , denoted by PQ . However, we are more interested in the directed line segment-- PQ which begins at with tail P and ends at head Q : d79 d15- Q-- PQ d55 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 d111 P d111 d47 | | | | | | | | R 2 Using this, we identify the directed line segment-- PQ with the 2-vector x = bracketleftbigg x - x y - y bracketrightbigg ....
View Full Document

Page1 / 4

lecture_16- Vector Spaces - MA 265 LECTURE NOTES:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online