Linear Maps Flashcards
What is a vector space?
A set acting on a field, that is closed under vector addition and scalar multiplication
What is a linear map?
A map between 2 vector spaces that respects addition and scalar multiplication
What is an isomorphism?
A linear map which is bijective (injective and subjective)
What does injective mean?
f(x) = f(y) if and only if x = y
What does surjective mean?
For all w in W, there exists a v in V such that f(v) = w
What can we say about a linear map inverse?
A linear maps inverse is also linear
If a map is injective, what is preserved?
Linear independence is preserved if a map is injective
If a map is surjective, what is preserved?
Spanning is preserved if a map is surjective
If a map is bijective, what is preserved?
Basis is preserved if a map is bijective (isomorphism)
What is the dimension of a vector space?
The dimension of its basis
When is V isomorphic to F^n?
When the dimension of V is n
If dim(V) = n, what do we know about the spanning set?
Has at least n vectors
If dim(V) = n, what do we know about the linear independent set
Has at most n vectors
If dim(V) = n, what do we know about the basis
Has exactly n vectors
If W is a subspace of V, what can we say about their dimensions?
dim(W) is less than or equal to dim(V). These are equal only if W = V.
What does GL_n(K) represent?
The set of invertible (nxn) matrices with coefficients in the field K.
What does it mean for a polynomial to annihilate a matrix?
A polynomial annihilates a matrix “M” if p(M) = 0
What is the characteristic equation?
For a matrix “A”, C_A(X) = det(XI - A)
What does the characteristic equation do to it’s own matrix?
The characteristic equation annihilates it’s own matrix, C_A(A) = det(AI - A) = 0
What does the matrix M(f, B, C) represent?
The matrix of “f” with respect to the basis B and C. Coefficients are given by the coefficient of f(B) = C.
What does it mean for matrices to be similar?
There exists an invertible matrix P such that
A’ = P^(-1) * A * P
What can we say about the characteristic equations of the map f and corresponding matrix A?
They are equal, C_f(X) = C_A(X).
What is an eigenvalue?
If f is a linear map, an eigenvalue (Z) is such that
f(v) = Zv. This is equivalent to “v” being in the nullspace of (f - ZI)
If an eigenvalue of f exists, what can we say about f’s nullspace?
The nullspace of “f - ZI” is not empty. and hence the eigenvalue is a root of C_f(X)
What is the kernel?
Ker(f) = {vectors in V : f(v) = 0} = null(f)
What is the kernel equivalent to?
The kernel of f is equivalent to the nullspace of f. This is a subspace of V.
What is the image?
Im(f) = {vectors in W : f(v) = w}. This is a subspace of W.
What is the nullity?
Dimension of f’s kernel, dim(Ker(f))
What is the rank?
Dimension of f’s image, dim(Im(f))
What is the rank-nullity formula?
For a linear map, dim(V) = rank(f) + nullity(f)
What is the minimal polynomial?
A monic polynomial, of the smallest possible degree, that annihilates a matrix
What is the maximum degree of the minimal polynomial?
Its degree is at most the degree of the characteristic equation of the matrix
What does x being a root of the minimum polynomial mean?
x is an eigenvalue
If a polynomial annihilates a matrix what do we know about the minimum polynomial?
The minimum polynomial divides the annihilating polynomial
What is the eigenspace of an eigenvalue “x”?
The nullspace of (A - xI) is called the eigenspace E(x). This consists of eigenvectors with eigenvalue x.
If A is an (nxn) diagonalizable matrix, what do we know about the eigenspace’s dimension?
The sum of dim(E(x)) where x represents all distinct eigenvalues is equal to n
What is Jordan Normal Form?
A matrix with eigenvalues on its leading diagonal, 0 or 1 in its super-diagonal, and 0 everywhere else.
How do we find the JNF of a matrix?
Find the characteristic equation, then refer to the case section!
When is a matrix diagonalizable?
If the minimum polynomial has simple roots (all eigenvalues are distinct), or if a polynomial annihilates the matrix.
