1. Definitions
1.1. Vector Spaces
 Let be a set, and be a field. We say is a vector space if there exists two binary operations and , under which is closed and satisfies:
 is a commutative group
 If , then

 , where is the identity of
 Subspaces are subsets of , which are closed under vector addition and scalar multiplication
 If be the smallest subspace of containing , is called the subspace generated by
 It can be shown that,
 finite dimensional: is finite dimensional over if a finite set , s.t. . Otherwise, is infinite dimensional.
 basis: is a basis of over if , and no proper subset of spans
 LI: is LI iff
is LI if every finite subset of is LI
 Complementary subspace: Let be a subspace of . We say a subspace of complementary of if
1.2. Quotient Space
 is a vector space over and is a subspace of . Quotient space
addition:
multiplication:
can be shown to be a vector space
1.3. Linear Mappings
 Let be two vector spaces over same field . A mapping is linear if
 is isomorphism if it is oneone, onto and linear
 Let be a vector space over . is called the dual space of
 Given a basis of , we can derive a basis of the dual space as
 For any , define, . Set of all such ‘s form the dual of and is denoted by
 Define
 is linear, oneone, in addition onto for finite dimensional vector spaces.
 Annihilator: Let be a vector space and , annihilator of is defined as,
 Let , linear. Let and be two bases respectively.
The matrix associated with this transformation is,
 Given , we associate, given by
1.4. Inner Product Spaces
 An orthonormal set in an inner product space is said to be complete, if this can’t be properly included in any other orthonormal set.
 For a given linear transformation , the transformation is called the adjoint of if
 We say is a positive operator if and (nonnegative semidefinite). Further, is strictly positive if is positive and
 Let be a unitary space (complex inner product), then which preserves inner product is called unitary transformation. If is Eucliedean (real inner product), is orthogonal transformation
 Let be an Eucliedean space and . We say is a rigid motion if . A special kind of rigid motion is translation, is called translation by . e.g., any unitary transformation is a rigid motion.
2. Theorems
2.1. Vector Spaces
 Every field is a vector space over itself
 Let be a finite dimensional vector space over , then has a basis
 If span and are LI, then it is a basis
 If span and , then it is a basis
 If is a finite dimensional vector space having a basis consisting of elements, any LI set of vectors in is a basis of
 Let . Then any two bases of have same number of elements
 If is a subspace of a finite dimensional vector space , , and equality iff
 For a finite dimensional vector space , the following statements are equivalent
 is a basis of
 is maximal linear independent set
 is minimal generating set
 Subspaces and are complementary iff each can be uniquely written as . We say , direct sum
 If is a subspace of a finite dimensional vector space , then has a complementary subspace

Theorem 1 Let and be two subspaces of a finite dimensional vector space , then
2.2. Quotient Space

Theorem 2 Let be a finite dimensional vector space over and is a subspace of . Then
2.3. Linear Mappings
 For any linear transformation , the vector spaces and are on same scalar field
 Let be linear and s.t. , then
 Let , be linear, then
 If are LI in and oneone, then are LI in
 Let , then is oneone

Theorem 3 : Let and be vector spaces. Let is a basis of . Let be any set of (not necessarily distinct) vectors in . Then a unique linear map such that

Theorem 4 (RankNullity Theorem) : Let be a finite dimensional vector space and , linear, then
 If be an isomorphism, so is
 , and finite dimensional and , then is 11 is onto

Theorem 5 (Fundamental Theorem of Homomorphism) : Let and be vector spaces over . be a surjective linear map. Then,
 Let and be vector spaces. Then the dimension of is
 If then
 If and be a subspace of , then
(The proof is useful for many similar proofs)
 Solution of homogeneous linear equations: A homogeneous linear system of equations has solution iff is in the annihilator of the row space of .
 Solution of nonhomogeneous linear equations: Let be a set of vectors in and be scalars. Let
then such that iff
In other words, nonhomogeneous system of equation has a solution iff the row rank of the coefficient matrix is equal to the row rank of its augmented matrix.
 If is the matrix associated with , that with will be
2.4. Inner Product Spaces
 Useful inequalities
CauchySchwartz:
Hölder’s: Let real and be s.t.This is a generalization for CauchySchwartz in ordinary real inner products
Minkowski’s: Let realnorm on

Theorem 6 Let be an orthogonal set not containing the zero element, then is LI

Theorem 7 Let be an inner product space and be an LI set. Then a set in s.t.
 is orthonormal
 for each ,
The proof of this uses the GramSchmidt orthogonalization procedure, which is useful for extracting orthogonal basis from LI. The orthogonal set of vectors can be written as,
Further we normalize it to get the orthonormal set .

Theorem 8 Let be an orthonormal basis for an inner product space , then for any
for any

Theorem 9 Let be a set in , s.t. . Suppose
then is an orthonormal basis

Theorem 10 (Bessel’s inequality) Let be an orthonormal set in an inner product space , then for any
 Let be an orthonormal set in an inner product space , then the following are equivalent:
 is complete
 for
 For each ,
 For any
 For any

Theorem 11 Let be a finite dimensional subspace of an inner product space , then . is the orthogonal complement of

Theorem 12 (Pythagoras) Let be a finite dimensional subspace of an inner product space , then for each ,
 ; Best approximation

Theorem 13 (Riesz Representation Theorem) If be a finite dimensional inner product space and , then a unique such that . So, we write . . This way every can be uniquely identified by a
 The mapping given by can be shown to be oneone, onto and conjugate linear.
 For polynomial inner product space, , with inner product , if the functional is defined as a evaluation of the polynomial at some point , i.e., , then the only possible functional is the zero functional, i.e.,

Theorem 14 Let and suppose exists, then
 If , and has same rank
 Some properties of the adjoint transformation:

Proposition 15 is selfadjoint iff the matrix of wrt some orthonormal basis of is selfadjoint (Hermitian)

Proposition 16 If is a complex inner product space and , ( not necessarily selfadjoint), then
This is a special property for complex, if , this is true for any inner product

Proposition 17 Let be a complex inner product space and , then

Proposition 18 Let be an inner product space and , strictly positive, then is invertible

Proposition 19 Let be an inner product space with , is an orthogonal projection iff
 Every orthogonal projection is a positive linear transformation

Theorem 20 Let preserves inner product, then
 preserves norm (isometric)
 is linear
 is oneone
 If , then is invertible

Proposition 21 is a unitary transformation on a finite dimensional inner product space if and only if maps orthonormal basis to orthonormal basis

Proposition 22 Let be unitary and , . Then,

Theorem 23 Let be a rigid motion (preserves distance in Euclidean space, real inner product), then , where is a translation by and is unitary
Basically,