In the next chapter, we will use the following
sets with elements called vectors.
The set Rn is the set of n-tuples ( x1,x2,x3,….,xn )
.
Thus, R is the set of real numbers;2,-1,1.2
are some of its elements or vectors.
R2 is the set of ordered pairs (x,y)….Some of
vectors are (1,-2),(0,0) and (2.12,-1.43)…..
R3 is the set of triples (x,y,z)……Some of its
vectors are (2,-1,0) and (-3,0,7) .
In these sets, addition of vectors is defined
as adding the corresponding elements……
Exs. 1. (2,-3)+(3,-4)=(2+3,-3-4)=(5,-7)…….
2. (0,0,-3)+(2,2,3)=(2,2,0)……..
Scalar multiplication is done by multiplying
each element by the scalar….
Exs.……
1.
2.
-3(2,2,-1)=(-6,-6,3).
2(-2,3)-3(1,-1)=(-4,6)-(3,-3)=(-7,9)……
Now, Pn is defined as the set of polynomials
with degree n or less………….
P2=set of polynomials with degrees 2 or
less…..
Exs…2t2-t+2;3t-4; and t2-t are some of
vectors in P2.
P3=set of polynomilas with degree 3 or
less…
Exs. t3-4t-3; t2+2; t3-1…are 3 of the vectors
in P3….
Addition and Scalar Multiplication on these sets
are the usual addition/scalar multiplication.
Exs.
1.
2.
2(t2-2)-4(t2-3t+1)
=2t2-4-4t2+12t-4
=-2t2+12t-8
-2(t3-2)+3(t3-t-1)+2(t2+4t)
=-2t3+4+3t3-3t-3+2t2+8t
=t3+2t2+5t+1.