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LINEAR ALGEBRA NOTES
8. Vector spaces
vector space: set V of vectors with verctor addition and scalar
multiplication satisfying
for all and
examples: ,P
polynomials, polynomials with degree less
than n, sequences,
sequences converging to 0, functions on R, C(R) continuous
functions on R, solutions of
homogeneous systems
subspace of V : subset W of V that is a vector space with same operations
proper subspace of V : subspace but not
and not V
examples:
W = and W = V , subspaces of V
W = lines through origin, subspace of
W =planes through origin, subspace of
W =diagonal n × n matrices, subspace of
W = , subspace of V where
W =convergent sequences, subspace of V =sequences
W =continuous functions on R, subspace of V =functions on R
fact: subset W of V is a subspace of V iff
nonempty:
closed under addition :
closed under scalar multiplication :
9. Linear independence
linearly independent:
implies
linearly dependent: not independent
parallel vectors: one is scalar multiple of the other
notation
properties :
u, v linearly independent
vectors are dependent i one of them is linear combination of the others
subset of lineraly independent set is linearly independent
columns of matrix A are independent i AX = 0 has only trivial solution
columns of square matrix A are independent i A invertible iff detA ≠ 0
independent,
implies
independent
independent,
implies
rows of row echelon matrix are independent
leading columns of echelon matrix are independent
10. Bases
S spans W: spanS = W
S is a spanning set of W
basis of V : linearly independent spanning set of V
maximal independent set in V
minimal spanning set of V
standard bases for V :
properties:
implies T dependent
all bases of V has same number of vectors
dimension of V : dimV =number of vectors in a basis of V
examples:
properties:
W proper subspace of V implies dimW < dimV
independent subset of V can be extended to a basis of V
spanning set of V contains a basis of V
11. row, column and null spaces
notation: sizeA = m× n
row space of A: RowA =subspace of
spanned by rows of A
row rank of A: dim RowA
column space of A: ColA =subspace of
spanned by columns of A
column rank of A: dim ColA
algorithm for basis of RowA:
(i) reduce A to echelon form B
(ii) take nonzero row vectors of B
algorithm for basis of ColA:
(i) reduce A to echelon form B
(ii) take columns of A corresponding to leading columns of B
fact: row rank A equals column rank A
rank A: this common value
null space of A: NullA = = solution
set of homogeneous system, subspace of
properties:
A, B row equivalent implies RowA = RowB
A, B row equivalent implies colums of A and columns of B have the same
dependence relations
Ax = b consistent iff b ∈ ColA
rankA + dim NullA = n
12. Coordinates
notation: bases for
eng standard basis for V
fact: each v ∈ V can be written uniquily as
coordinates of v in basis B:
huge fact: is an isomorphism (
are the 'only' finite dimensional vector spaces)
transition matrix from basis B to basis D:
square matrix
properties:
algorithm for finding a basis for
in V :
(i) find a bases B for V (use standard if possible)
(ii) put the coordinates of the v_{i}'s as rows (columns) for a matrix A
(iii) find a basis for the rowspace (columnspace) of A
(iv) use this basis as coordinates to build the basis of W
13. Linear transformations
notation: basis for
basis for W, E standard basis for V
linear transformation: L : V → W such that for all
L(u + v) = L(u) + L(v) additive
multiplicative
kernel:
range:
L is onetoone (11): L(u) = L(v) implies u = v
L is onto W: ranL = W
properties:
kerL subspace of V
ranL subspace of W
L is 11 i
matrix of L:
properties:
R, S are similar matrices: S = P^{1}RP for some P
fact: R, S are similar iff
for some L : V → V and bases B, D for V
(P is the transition matrix)
rank of L: rankL = dim ranL
properties:
[ranL]D = ColM
[kerL]B = NullM
rankL = rankM
dim kerL = dim nullM
rankL + dim kerL = dimV
14. Eigenvalues and eigenvectors
notation: L : V → V linear transformation,
coordinates of u
eigenvalue problem:
transformation version
eigenvalue: λ
eigenvector of L associated to λ: u
eigenspace associated to λ:
matrix version
eigenvalue: λ
eigenvector of A associated to λ: x
eigenspace associated to λ :
characteristic polynomial:
if A~ B then charpoly(A) = charpoly(B)
characteristic equation : λ eigenvalue of A iff
15. Diagonalization
A diagonalizable: A similar to diagonal matrix
fact: implies
is a basis of eigenvectors with associated
eigenvalues in the diagonal od D
properties:
if eigenvectors associated to distinct
eigenvalues then they are independent
if sizeA = n × n and A has n distinct eigenvalues then A diagonalizable
distinct eigenvalues,
bases for eigenspaces implies
is independent
algorithm for diagonalization:
(i) solve charachteristic equation to find eigenvalues
(ii) for each eigenvalue nd basis of associated eigenspace
(iii) if the union of the bases is not a basis for the vectorspace than not
diagonalizable
(iv) build P from the eigenvectors as columns
(v) build D from the corresponding eigenvalues
16. Bilinear functional
product of U and V :
bilinear functional on V : such that
for all
and
fact: Every bilinear functional f on
is
for some
where
The bilinear functional f can be
symmetric: f(u, v) = f(v, u) for all
positive semi definite: f(v, v) ≥0 for all v ∈ V
positive definite: f(v, v) > 0 for all
negative semi definite: f(v, v) ≤0 for all v∈ V
negative definite: f(v, v) < 0 for all
indefinite: neither positive nor negative semidefinite
17. Inner product
inner product: symmetric, positive definite, bilinear functional
examples of inner products:
dot product (standard inner product) on
:
standard inner product on C[0, 1]: (continuous functions on [0,1]),
fact: every inner product on
is
where A is a symmetric (therefore
diagonalizable)
matrix with positive eigenvalues and
length (norm):
unit vector:
unit vector in the direction of v:
distance:
angle:
orthogonal: iff
iff
orthogonal:
for all i, j
fact: nonzero orthogonal vectors are independent
orthonormal: S is orthogonal and
for all i
Cauchy Schwartz inequality :
Triangle inequality:
Pythagorean theorem:
implies
orthogonal complement: , W is subspace
of V
properties: W is subspace of
is a subspace
W = span(S), for all i implies
(basis of W) [ (basis of
) is basis of
18. Orthogonal bases and GramSchmidt algorithm
fact: orthogonal basis for a
subspace W of V , y ∈ V
if such that
and
then and
orthogonal projection: = the unique
p ∈ W such that
GramSchmidt algorithm: for finding an orthogonal basis
for
(i) make independent if necessary
(ii) let
(ii) inductively let
fact:
19. Least square solution and linear regression
fact: if W subspace of V , w ∈ W, y ∈ V then
is minimum when
fact: is minimum
least square regression line ax + b: data
, β makes
minimum, that is,
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