Linear Maps

Question 1

What is a vector space?

Answer 1

A set acting on a field, that is closed under vector addition and scalar multiplication

Question 2

What is a linear map?

Answer 2

A map between 2 vector spaces that respects addition and scalar multiplication

Question 3

What is an isomorphism?

Answer 3

A linear map which is bijective (injective and subjective)

Question 4

What does injective mean?

Answer 4

f(x) = f(y) if and only if x = y

Question 5

What does surjective mean?

Answer 5

For all w in W, there exists a v in V such that f(v) = w

Question 6

What can we say about a linear map inverse?

Answer 6

A linear maps inverse is also linear

Question 7

If a map is injective, what is preserved?

Answer 7

Linear independence is preserved if a map is injective

Question 8

If a map is surjective, what is preserved?

Answer 8

Spanning is preserved if a map is surjective

Question 9

If a map is bijective, what is preserved?

Answer 9

Basis is preserved if a map is bijective (isomorphism)

Question 10

What is the dimension of a vector space?

Answer 10

The dimension of its basis

Question 11

When is V isomorphic to F^n?

Answer 11

When the dimension of V is n

Question 12

If dim(V) = n, what do we know about the spanning set?

Answer 12

Has at least n vectors

Question 13

If dim(V) = n, what do we know about the linear independent set

Answer 13

Has at most n vectors

Question 14

If dim(V) = n, what do we know about the basis

Answer 14

Has exactly n vectors

Question 15

If W is a subspace of V, what can we say about their dimensions?

Answer 15

dim(W) is less than or equal to dim(V). These are equal only if W = V.

Question 16

What does GL_n(K) represent?

Answer 16

The set of invertible (nxn) matrices with coefficients in the field K.

Question 17

What does it mean for a polynomial to annihilate a matrix?

Answer 17

A polynomial annihilates a matrix “M” if p(M) = 0

Question 18

What is the characteristic equation?

Answer 18

For a matrix “A”, C_A(X) = det(XI - A)

Question 19

What does the characteristic equation do to it’s own matrix?

Answer 19

The characteristic equation annihilates it’s own matrix, C_A(A) = det(AI - A) = 0

Question 20

What does the matrix M(f, B, C) represent?

Answer 20

The matrix of “f” with respect to the basis B and C. Coefficients are given by the coefficient of f(B) = C.

Question 21

What does it mean for matrices to be similar?

Answer 21

There exists an invertible matrix P such that

A’ = P^(-1) * A * P

Question 22

What can we say about the characteristic equations of the map f and corresponding matrix A?

Answer 22

They are equal, C_f(X) = C_A(X).

Question 23

What is an eigenvalue?

Answer 23

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)

Question 24

If an eigenvalue of f exists, what can we say about f’s nullspace?

Answer 24

The nullspace of “f - ZI” is not empty. and hence the eigenvalue is a root of C_f(X)

Question 25

What is the kernel?

Answer 25

Ker(f) = {vectors in V : f(v) = 0} = null(f)

Question 26

What is the kernel equivalent to?

Answer 26

The kernel of f is equivalent to the nullspace of f. This is a subspace of V.

Question 27

What is the image?

Answer 27

Im(f) = {vectors in W : f(v) = w}. This is a subspace of W.

Question 28

What is the nullity?

Answer 28

Dimension of f’s kernel, dim(Ker(f))

Question 29

What is the rank?

Answer 29

Dimension of f’s image, dim(Im(f))

Question 30

What is the rank-nullity formula?

Answer 30

For a linear map, dim(V) = rank(f) + nullity(f)

Question 31

What is the minimal polynomial?

Answer 31

A monic polynomial, of the smallest possible degree, that annihilates a matrix

Question 32

What is the maximum degree of the minimal polynomial?

Answer 32

Its degree is at most the degree of the characteristic equation of the matrix

Question 33

What does x being a root of the minimum polynomial mean?

Answer 33

x is an eigenvalue

Question 34

If a polynomial annihilates a matrix what do we know about the minimum polynomial?

Answer 34

The minimum polynomial divides the annihilating polynomial

Question 35

What is the eigenspace of an eigenvalue “x”?

Answer 35

The nullspace of (A - xI) is called the eigenspace E(x). This consists of eigenvectors with eigenvalue x.

Question 36

If A is an (nxn) diagonalizable matrix, what do we know about the eigenspace’s dimension?

Answer 36

The sum of dim(E(x)) where x represents all distinct eigenvalues is equal to n

Question 37

What is Jordan Normal Form?

Answer 37

A matrix with eigenvalues on its leading diagonal, 0 or 1 in its super-diagonal, and 0 everywhere else.

Question 38

How do we find the JNF of a matrix?

Answer 38

Find the characteristic equation, then refer to the case section!

Question 39

When is a matrix diagonalizable?

Answer 39

If the minimum polynomial has simple roots (all eigenvalues are distinct), or if a polynomial annihilates the matrix.