Math 343 Review Sheet
Chapter 2
1. Definition of and how to find an echelon form of a matrix. Also, the reduced row echelon form
(RREF).
2. Apply REF and RREF to solving linear systems.
3. Know how to determine whether a linear system is consistent. If so, know when it has only one or
infinitely many solutions. In the latter case, you should be able to parameterize the solutions and find the
dimension and bases for the solution space (see Chapter 4 below).
4. Definition of and what types of elementary matrices exist.
5. How to find the inverse of a square matrix.
CHAPTER 3
1. Definition of determinant. Know how to compute 2x2 and 3x3 determinants.
2. Know properties of determinants (as in Section 2) such as how they change when two rows/columns
are swapped, or when a multiple of a row/column is added to another row/column, or when one
row/column is multiplied by a scalar.
3. Know what cofactors are. Know how to compute determinants by using cofactor expansion. Know how
to compute areas of parallelograms or triangles by using determinants.
4. Know the relation between determinants and linear independence/dependence of vectors.
CHAPTER 4
1. Know the definition of a real vector space (Section 2). Be able to check that a given set with two given
operations is a vector space.
2. Know what a vector subspace of a vector space is. Should be able to check that a given subset of a
vector space is a subspace (as in Section 3). Be familiar with equations (cartesian and/or parametric) of
lines and planes in the 3-dimensional space. Know how to find the equation of a line through two given
points, and the equation of a plane through three given points. Know how to switch between cartesian and
parametric equations.
3. Definition of a spanning set for a subspace of a vector space (including the space itself). Should know
how to check that a given vector (in some Rn or some polynomial space Pn(X)) lies in the span of a given
set of vectors. Should be able to find a spanning set for a given subspace.
4. Know (and understand!) the definition of linear independence. Should be fluent in showing that a given
set of vectors is l.i. or linearly dependent (l.d.). Know what a maximal l.i. subset of a given set of vectors
is, and know how to find it. Should also be able to find a minimal l.d. subset of a given set of vectors.
5. Know everything (definitions, theorems) from Section 6, perhaps the most important section we have
studied! Should be able to complete an l.i. set of vectors to a basis. Also, to extract a basis from a
spanning set.
6. Know what coordinates of a vector w.r.t. a basis are. Be able to find them. Should know how to find
the transition matrix between two bases and, thus, if you know the coordinates of a vector in one basis,
then you should know how to find them in the other basis.
7. Know what the rank and nullity of a matrix are. Be able to find them by REF. Be familiar with the
connection between them, and their connection with the singularity or non-singularity of square matrices.
CHAPTER 5
1. Know the definition of the inner-product on a vector space V. Be able to check that a function defined
on VxV is an inner-product (especially the condition (u,u)=0 iff u=0, which most of you had trouble with
on Test 3). Know what length of a vector is and how to compute it (depending on the inner-product).
Know what orthogonality means. Know the connection between orthogonality and linear independence.
2. Know what the matrix of an inner-product w.r.t. an ordered basis is, i.e.know that given an ordered
basis S for a Euclidean space (know what that means), then any inner-product can be written as
(u,v)=[u]ST C[v]S for a positive definite matrix (know what that means and how to check that; no one who
tried Problem 6 could check that on Test 3!)
3. Know how to construct by the Gram-Schmidt process an orthonormal basis (know what that means)
basis for a given subspace of Rn (could be given as the column space of some nxm matrix, the row space
of an mxn matrix or, for example, as the space of vectors x=[a,b,c,d] such that b+d=0, etc.).