lecture_16- Vector Spaces

# lecture_16- Vector Spaces - MA 265 LECTURE NOTES WEDNESDAY...

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

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