Lecture Chapter 4 Flashcards
What is a vector space?
nonempty set of vectors such that
u+v is in V
cu is in V
What is a subspace of a vector space?
subset H of V with 3 properties
- zero vector is in H
- H is closed under vector addition : u+v is in H
- H is closed under multiplication by scalars : cu is in H
If v1,v2,…vp are in a vector space V, what about Span {v1,v2,…,vp}?
Span{v1,v2,…,vp} is a subspace of V
What is a null space?
Nul A : set of all solutions of homogeneous equation Ax=0
A mxn
Nul A : subspace of Rn
Col A : subspace of Rm
What is a column space?
Col A : set of all linear combinations of the columns of A
Col A = Span {a1, a2, … , an}
Is Col A all Rm?
only if Ax=b has a solution for every b in Rm
What is a linear transformation T from a vector space V into a vector space W
assigns to each x in V a unique T(x) in W such that
- T(u+v) = T(u) + T(v)
- T(cu) = cT(u)
What is the kernel of a linear transformation?
the set of all u in V such that T(u)=0
What is the range of a linear transformation?
the set of all vectors in W (T(x)) for some x in V
When is an indexed set {v1, v2, … , vp} with v1/=0 linearly dependent?
if some vj is a linear combination of the preceding vectors v1, v2,…, vj-1
H subspace of vector space V
When is B = {b1, b2,…,bp} a basis for H?
B is a linearly independent set
the subspace spanned by B coincides with H : H=Span{b1,b2,…,bp}
What is the spanning set theorem?
S={v1,v2,…,vp} a set in V , H=Span{v1,v2,…,vp}
- if one of the vectors in S is a linear combination of the remaining ones, the set formed by S - this vector still spans H
- if H /= {0}, some subset of S is a basis for H
B a basis for V and x in V
What are the B-coordinates of x?
x = c1b1 + c2b2 + … + cnbn
What is the coordinate vector of x relative to B
[x]B=(c1,c2,…,cn)
mapping x–>[x]B is the coordinate mapping
What is the unique representation theorem
there exists a unique set of scalars such that x=c1b1 + c2b2 + … + cnbn
What is the change of coordinates matrix from basis B to standard basis in Rn?
PB = [b1 b2 … bn]
x=PB[x]B
PB^-1 x = [x]B
What is a characteristic of the coordinate mapping x–>[x]B
it is a one to one linear transformation from V onto Rn
What is isomorphism
one-to-one linear transformation from a vector space onto another
Properties about isomorphism
- a vector space V with basis B, containing n vectors is isomorphic to Rn
- Pn and Rn+1 are isomorphic
- n : dimension of vector space V
Vector space V, basis B {b1, b2, … , bn}
What if a set in V contains more than n vectors
the set is linearly dependent
What if a vector space V has a basis of n vectors?
then every basis of V must have n vectors
What about a vector space V spanned by a finite set?
V is finite-dimensional
dim V is the number of vectors in a basis for V
What is the dimension of the zero vector space {0}?
zero
V finite dimensional
H subspace of V
What can you say?
any linearly independent set in H can be expanded to a basis for H
H is finite-dimensional
dim H <= dim V
What is the basis theorem?
V : p-dimensional, p>=1
- any linearly independent set of exactly p elements in V is automatically a basis for V
- any set of exactly p elements that spans V = basis for V
How can you find dim Nul A?
number of free variables in Ax=0
How can you find dim Col A?
number of pivot columns in A
What is the row space of A?
Row A : set of all linear combinations of row vectors
subspace of Rn
Row A = Col A^T
If A and B are row equivalent, what can you say about their row space?
they have the same row space
-B in echelon form : nonzero rows of B form a basis for row space of A and B
What is the rank of A?
dimension of the column space of A
What is the rank theorem?
dim Col A = dim Row A
rank A = number of pivot positions in A
rank A + dim Nul A = n
Invertible Matrix Theorem continued
- columns of A form a basis for Rn
- Col A = Rn
- dim Col A = n
- rank A = n
- Nul A = {0}
- dim Nul A = 0
