mathematical structureformed by collection of vectors. the operations of vector addition and scalar multiplication have satisfy 10 axioms

Axioms1. u and v is in V, so u+v is in V2. u+v=v+u3. u+(v+w)=(u+v)+w4. There is 0 in V & it is called zero vector for V, such that 0+u=u+0=u for all u in V5. For each u in V, there is an object -u in V, such that u+(-u)=(-u)+u=06. k is any scalar and u is any object in V, then ku is in V7. k(u+v)=ku+kv8. (k+m)u=km+mu9. k(mu)=(km)(u)10. 1u=u

If W is itself a vector space under the condition and scalar multiplication defined on V

THEOREM 4.2.1 If W is a nonempty set of vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisfied.1. If u and v are vectors in W, then u + v is in W.2. If k is a scalar and u is a vector in W, then ku is in W.

Axioms 1,4,5,6 under addition and scaar multiplicationIf axioms 1 and 6 are hold in subspace----->real subspacesubspace of R2-----> line through the originsubspace of R3-----> (line through the origin) (planes through the origin)

THEOREM 4.2.3 The solution set of a homogeneous system Ax = 0 of m equations in n unknowns is a subspace of Rn.

If w is a vector in a vector space V, then w is said to be a linearcombination of the vector v1, v2, …, vr in V if w can be expressed in the form: w =k1v1 + k2v2 +...+ krvrk = the coefficients of the linear combination

If vector is spanned, then its subset is also spanned

A set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent

If is S is a set of vectors in a finite-dimensionalvector space V, then S is called abasis for V if:(a) S spans V.(b) S is linearly independent.

Standard basis for R2 ===> (1,0), (0,1)Standard basis for R3 ===>(1,0,0 ), ( 0,1,0), (0,0,1)Standard basis for P3 ===> S =(1, x, x^2 , x^3)Standard basis for M2x2

THEOREM 4.5.1 Uniqueness of Basis RepresentationIf S = {v1, v2, …, vn} is a basis for a vector space V, then every vector v in V can be expressed in the form v = c1v1 + c2v2 + ⋯ + cnvn in exactly one way.

THEOREM 4.6.1All bases for a finite-dimensional vector space have the same number of vectors.

THEOREM 4.6.2Let V be a finite-dimensional vector space, and let (v1+...+vn) be any basis for V.If a set has more than n vectors, then it is linearly dependent.If a set has fewer than n vectors, then it does not span V.

Real numberPolynomialsMatrix

If V is a vector space with basis {v1, v2,..., vn}, then every vector v ∈ V can be written uniquely as a linear combination of v1, v2,..., vn.

Form Matrix (old/new)

Transition Matrix

General solution for Ax=B

Row Space

Column Space

Null Space

Rank

Nullity

Fundamental Spaces