Review Topics for Final Exam

This is a compilation of the topics for the first two in-class exams, together
with a few additional topics . The new topics are in boldface.

1 linear equations

1.1 Systems of linear equations

Know how to solve a linear system over a field F by forming the augmented
matrix and applying row ope rations .

1.2 Row reduction and echelon forms

Know how to transform a matrix in the row-reduced echelon form.
What is the relation between elementary matrices and elementary
row operations ? Does the solution set to a linear system change under elementary
row operations? What are independent/free variables? How can we
tell there are free variables by looking at the row-reduced echelon form?

1.3 Vector equations

How can we rewrite a linear system Ax = b in vector form? Can we solve the
system if b can be written as a linear combination of the column vectors of A?

1.4 Solution sets

How do the solutions to an inhomogeneous system relate to the solutions of the
corresponding homogeneous one?

1.5 Matrix inverse

Know how to compute the inverse of a matrix. What is the inverse of AB in
terms of A^-1 and B^-1? What are the properties of the system Ax = b if A is
invertible? How can you use the determinant to test whether a matrix
is invertible?

2 Vector spaces

What are the defining properties of a vector space? Know the difference between
finite and infinite-dimensional vector spaces.

2.1 Subspaces

Know how to test whether a subset of a vector space is a subspace.

2.2 Spanning sets and linear independence

What is the span of a set of vectors? When are vectors linearly independent?

Does the span of the row vectors of a matrix A change under elementary
row operations? What about the span of column vectors?

If Ax = 0 has non-trivial solutions, what can we say about the column
vectors of A?

2.3 Bases and dimension

You can always extend a linearly independent set to a basis by adding appropriate
vectors to it. What condition is needed when adding a vector to a linearly
independent set to preserve linear independence?

Compute the dimension of W₁ +W₂ for subspaces W₁ and W₂. How many
vectors are needed for a spanning set, how many can a linearly independent set
have?

2.4 Coordinates and change of coordinates

How do we compute the change of coordinates matrix? How do we compute
coordinates of a vector with respect to a new basis?

3 Linear Transformations

Tell the difference between linear and non-linear transformations. Know how to
compare two linear transformations efficiently by invoking a basis.

3.1 Kernel and range, nullity and rank

What is the sum of nullity and rank? Describe the properties of a linear transformation
with zero rank or with zero nullity.

3.2 One-to-one, onto and invertibility properties

What do we need to check to verify a linear transformation is 1-1, onto, or
bijective? What does it mean if it is invertible? If T : V -> W and the
dimension of V equals that of W, what can we say about a 1-1 transformation?

What about a transformation which is onto? (Knowing rank nullity will be
helpful here.)

3.3 Isomorphism

What is an isomorphism? What properties does it have (think spanning sets,
linear independent sets, and bases)? If the vector space V is finite dimensional
and T : V -> W is an isomorphism, what about the dimension of W?

3.4 Matrix representation

How is it defined? Compare with change of coordinates. Coincidence? Know
how to convert statements for the matrix representation to statements about
the linear transformation. Matrix representation of the inverse? Know about
similar matrices and their role for deciding whether two matrices represent the
same linear transformation (with respect to different bases ).

3.5 Linear functionals

What is the dual space? What is the dual of a basis? What is the annihilator?
Relate dimension of subspace to that of its annihilator. Annihilator, sum and
intersection of subspaces.

3.6 Double dual

How can we map a vector space to its double dual? Is this always an isomorphism?

3.7 Transpose

Matrix representation for the transpose. Given T : V -> W and the coordinate
vector for a linear functional f with respect to a basis for W*, compute coordinates
of T^tf. Know how to relate between range and kernel of T and T^t with
the annihilator.

4 Eigen values and vectors

4.1 n-linear functions

Direct sums, n-linear functions as functions of matrices. Alternating n-linear
functions. If is n-linear and alternating, what is
for vectors

4.2 Determinant

Know how to express the determinant by co factor expansion and by
a formula involving permutations of indices. If a matrix contains
lots of zero entries, which method would you use to compute the
determinant? Does the determinant change under elementary row or
column operations? Can you use this to compute determinants?

4.3 Characteristic polynomial

How is it defined? What are eigenvalues?

4.4 Diagonalization

What are eigenvectors? Know how to find eigenvalues and eigenvectors.
Why is it nice if a matrix has a basis of eigenvectors? Which
steps do you need to fol low to diagonalize a matrix? What can prevent
a matrix from being diagonalizable?