: Vector Spaces, Span and Basis, Null Space, Eigen Vectors and Inner Product Flashcards
How do you calculate the determinant of a
a b
c d
matrix?
det(A) = ad-bc
What does the determinant indicate?
If determinant(A) is 0 then the matrix is not invertible
How do you calculate the determinant of a bigger matrix?
bring into 3x3 matrix and into echelon form and multiply the leading entries
What is a vector space?
A vector space is a non-empty set V of objects called vectors on which are defined two different operations. Addition and multiplication. All vectors have to be subjects to 10 axioms
What are the ten axioms of a vector space?
u,v, w are vectors in V and c and d are sclalars
- the cum of u and v (u+v) is in V
- u+ v = v + u
- (u + v) + w = u+ (v + w)
- there is a zero vector in V such that u + (-u) = 0
- For each u in V there is a vector -u such that u + (-u) = 0
What is the definition of a subspace?
a subspace of a vector space V is a subset H of V that has three properties
- the zero vector of V is in H
- H is closed under vector addition that is for each u and v in H the sum of u and v is in H
- H is closed under multiplication by scalars that is for each u in H and each scalars c the vector cu is in H
How is linear dependence defined?
vectors v1, …, vn are linearly dependent if the zero vector can be written as a nontrivial combinations of the vectors
0 = a1v1 + … + anvn
When is a set of vectors linearly independent?
if only the trivial solution(all 0 weights) is possible to achieve the zero vector
What is a clear indication that a set of vectors is linearly dependent?
- If the related matrix has more columns than rows
2. when the set S contains the zero vector
What is the null space of an m x n matrix?
Nul(A) is the set of all solutions of the homogenous equation Ax = 0.
What is the span of a set of vectors?
The span of a set of vectors is the set of all their linear combinations
i.e. vectors v and w
span = av + bw
Hence, the span of most 2d vectors is all vectors in 2d space but when they line up their span is all vectors whose tip sits on a line
in 3d:
span is flat sheet cutting through origin
What is the relationship between a vector sets span and its linear dependence?
If one of these vectors is redundant ( not adding anything to the span) then they are lineraly dependen
= one vector could be expressed as a linear combination of the others since it’s already in the span
What is the basis of a vector space defined like?
The basis of a vector space is the set of linearly independent vectors that span the full space
What is the span a single nonzero vector v?
span {v} = av : a in R
what is the span of the empty set?
Just the orgin
