Transcript modulo cv

6.1
Vector Spaces and Subspaces
Definition
Let V be a set on which two operations, called addition
and scalar multiplication, have been defined. V is called a
vector space if the following axioms hold for all vectors u,
v, and w in V and all scalars (real numbers) c and d.
1) u + v is in V
2) u + v = v + u
3) (u + v) + w = u + (v + w)
4) There exists an element 0 in V such that v + 0 = v
5) For each v in V, there is an element –v in V such that
v + (-v) = 0.
6) cv is in V
7) c(u + v) = cu + cv
8) (c + d)v = cv + dv
9) c(dv) = (cd)v
10) 1v = v
Examples
1) The set of m x n matrices with matrix addition
and scalar multiplication is a vector space.
2) Z3 = {0, 1, 2} with special operations (integer modulo 3)
is a vector space.
3) Let Σ be the set of all sequences, with the following
addition and multiplication:
If
a  a1 , a2 ,...
then
a  b  a1  b1 , a2  b2 ,...
b  b1 , b2 ,...
ka  {ka1 , ka2 ,...}
 Σ is a vector space.
Examples
1) The set of all real-coefficient polynomials of degree
three, together with usual addition and multiplication,
is not a vector space.
2) The set of all rational numbers with standard addition
and multiplication is a not vector space. However, the set
of all real numbers (complex numbers), with standard
addition and scalar multiplication, is a vector space
3) Z3 (the set of all vectors in three-dimensional
space whose components are integers), with usual
addition and multiplication is not a vector space.
Theorem
1)
2)
3)
4)
0v=0
c0 = 0
(-1)v = -v
If cv= 0 then c = 0 or v = 0
Examples and Notation
The following sets, with standard addition and scalar
multiplication, are vector spaces:
1) M = {M / M is a matrix}
2) P = {P / P is a polynomial}
3) F = {f(x) / f(x) is a function}
4) Mnxn = {M / M is an nxn square matrix}
5) Pn = {P / P is a polynomial of degree at most n}
6) D = {f(x) / f(x) is a differentiable functions}
Definitions
A subset W of a vector space V is called a subspace if W
itself is a vector space with the same scalars, addition
and scalar multiplications as V.
Zero space
Every vector space V has two subspaces: {0} and V.
Let W be a subset of a vector space V. Then W is a
subspace of V iff the following two conditions hold:
1) If u, v are in W, then u  v is in W.
2) If u is in W and c is a scalar, then cu is in W.
Examples
Show that:
1) W = {A/ A is a 3x3 symmetric matrix} is a subspace of M .
2) W = { f(x) / f(x) is a solution to the equation 2y’ + 4y = 0}
is a subspace of F .
3) W = {bx + cx2 / b,c are real numbers} is a subspace of P2 .
4) W = { 5 + bx + cx2 / b,c are real numbers} is NOT a
subspace of P2 .
5) W = { f(x) / f(x) is a solution to the equation 2y’ + 4y – 7= 0}
is NOT a subspace of F .
6) W = {an / an is a convergent sequence} is a subspace of Σ .
6.2
Linear Independence, Basis,
and Dimension
Terminology for a Vector Space V
 Linear combination:
A vector v is called a linear combinatio n of vectors
v1 , v 2 ,..., v n if there are scalars c1 , c2 ,..., cn such that
v  c1v1  c2 v 2  ...  cn v n .
For a set B  v1, v2 ,..., vn 
 Linear Dependence: There is one vector in B that can be
written as a linear combination of the other vectors in B.
 Linear Independence:
If c1v1  c2v2  ...  cn vn  0 , then c1  c2  ...  cn  0
 Spanning set: any vectors in V can be written as linear
combination of vectors in B.
 Basis: B is a linearly independent, spanning set for V.
 Dimension: dim V = number of vectors in a basis for V.
 A vector V is finite-dimensional if dim V is finite.
Examples
1) Is this set { x, 2x - x2 , 3x + 2x2 } linear independent in P2 ?
2) Is this set { sin 2x, sin x, cos x } linear independent?
3) Show that:
a) B = { x, 1 + x, x – x2 } is a basis for P2 .
b) B’ = { 1, x, x2 ,…, xn } is a basis for Pn .
c) Express 2 + 3x – x2 as a linear combination of vectors
in B and B’ .
Note: This is a unique representation with respect to
each basis.
Definition
Let B  v1 , v 2 ,..., v n  be a basis for a vector space V.
Let v be a vector in V, and write v  c1v1  c2 v 2  ...  cn v n .
Then c1 , c2 ,..., cn are called coordinate s of v with respect to B,
and the column vector v B
 c1 
c 
  2
:
 
cn 
is called the coordinate vector of v with respect to B.
u  vB  uB  vB
cuB  cuB
Theorems
Let V be a vector space with dim V = n. Then:
1. Any linearly independent set in V contains at most n
vectors, and can be extended to a basis for V.
2. Any spanning set for V contains at least n vector, and
can be reduced to a basis for V.
3. Any linearly independent set, or spanning set consisting
of exactly n vectors is a basis for V.
Examples
1) Is S = { x + 2, x – 2 } a basis for P2 ?
2) Extend S to a basis for P2 .
3) Find bases for the following vector spaces:
a) W1 = { 1 + 2ax + 3bx2 – 4bx3 } is a basis for P2 .
n
b) W2 = { [1, 2a, 3b, -4b] } is a basis for R .
c) W3 =  1
2a 
3b  4b


is a basis for M2x2
6.3
Change of Basis
Examples
1 
 3
Let B  v1 , v2  be a basis for R where v1    and v2   
 2
1
2
1
7
Let C  u1 , u 2  be a basis for R where u1    and u 2   
0
 8
2
1) Find coordinate vector of x = [-3, 8] with respect to B.
2) Find coordinate vector of x = [-3, 8] with respect to C.
3) Find a matrix P such that P [x]B = [x]C
Definition
Let B  u1 , u 2 ,..., u n  and C  v1 , v 2 ,..., v n  be bases for
a vector space V. The change - of - basis matrix from B to C
is denoted by PCB and given by :
PCB   u1 C , u 2 C , ... , u n C
