: 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
What is a span of three vectors?
When they are linearly independent: A plane in three dimensions
What is the dimension of NulA?
What is the dimension of ColA?
The dimension of NulA is the number of free variables in the equation Ax = 0
The dimension of colA is the number of pivot columns in A
What is the rank?
Rank A is the dimension of columns space of A (=number of pivot columns)
What is the eigenvalue of an eigenvector?
the factor by which it is stretched during the transformation
Can eigenvalues be negative?
Yes
What does the formula Av = lambda v imply?
v being the eigenvector it says that the matrix eigenvector multiplication equals the result of the mutiplication if the eigenvector and its eigenvalue
How do you calculate the eigenvector?
(A - lamda*identitymatrix) * vector = 0
What is an eigenvector?
an eigenvector of a nxn matrix A is a nonzero vector such that Ax = lambda x for some scalar lambda.
Lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax = lambda X such an
Can the zero vector be an eigenvector?
No an eigenvector must be nonzero
Can an eigenvalue be zero?
yes
What is the length or norm of v?
squareroot of the dot product of v
When are two vectors orthogonal?
When their innerproduct( dotproduct) = 0
