Math 307, Section D - Study Guide: Exam #1
1. Systems of Linear Equations:
• Know how to determine what type of solution (i.e. one unique, no solution, or infinitely
many solutions) will arise from a given linear system. Example:
3x1 + 2x2 − x3 = 5
7x1 − 2x2 + 3x3 = 4
• We also covered how to form an augmented matrix and used elementary row operations
on the matrix to determine the linear system’s solution. Augmented matrix for system
above:
3 2 −1 5
7 −2 3 4
• Make sure you understand the difference between the Row Echelon Form(REF) and the
Reduced Row Echelon Form(RREF) of a matrix. I will take off points if you use incorrect
terminology.
• Know how to find the parametric form for a set of infinite solutions. When you specify
the solution, free variables are the only permissible parameters for the basic variables.
2. Vectors:
• Know the vector addition and scalar multiplication rules, and understand the geometric
description of each operation for R2 .
• Know how to determine if a vector v is contained in the spanning set of a collections of
vectors (i.e Is v ∈ span{u1 , u2 , . . . , un }?). If it is, know how to determine the weights
c1 , c2 , . . . , cn which specify the linear combination that generates v (i.e. Find ci ,
1 ≤ i ≤ n, such that v = c1 u1 + c2 u2 , . . . , cn un ).
3. The Matrix Equation Ax = b:
• Know how to compute the product of a matrix and a vector.
• Given a matrix A ∈ Rm×n with column vectors a1 , a2 , . . . , an and the vector b ∈ Rm ,
the matrix equation Ax = b has a solution iff b ∈ span{a1 , a2 , . . . , an }.
• The homogeneous linear system Ax = 0 always has the trivial solution x = 0. This
system has nontrivial solutions iff the equation has at least one free variable. The
existence or nonexistence of such a solution has many many important implications.
Make sure you understand ALL of the the information about the linear transformation
T (x) = Ax that one gains from this system.
• Make sure that you can also identify if the solution to the system is a point, a line, or a
(hyper)plane.
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4. Linear Independence:
• A set of vectors {v1 , v2 , . . . , vp } are linearly independent if the vector equation
x1 v1 + x2 v2 + . . . + xp vp = 0
has only the trivial solution. The set of vectors is said to be linearly dependent if a
nontrivial solution exists.
• The above implies that the columns of the matrix A = [a1 a2 . . . an ] ∈ Rm×n are linearly
independent if the the matrix equation Ax = 0 has only the trivial solution x = 0.
• Note 1: If a set of vectors S = {v1 , v2 , . . . , vp }, with vi ∈ Rn , contains more vectors than
there are entries in each vector (i.e. p > n), then the set of vectors is linearly dependent.
• Note 2: If 0 ∈ Rn is an element of the set S, then the set of vectors is linearly dependent.
5. Linear Transformations:
• Make sure that you know the definitions of domain, co-domain, image, and range for a
linear transformation.
• A transformation T : Rm → Rn is linear if:
(a) T (u + v) = T (u) + T (v) for all u, v in the domain of T
(b) T (cu) = cT (u) for c ∈ R and all u in the domain of T .
• The following is typically how one checks for linearity. If a transformation T : Rm → Rn
is linear, then for all u, v in the domain of T and all scalars c, d ∈ R, it must satisfy:
(a) T (0) = 0
(b) T (cu + dv) = cT (u) + dT (v)
• A transformation T : Rm → Rn defined by
T (x) = Ax for all x ∈ Rn
is called a matrix transformation and will always be linear (Make sure you know how to
show this).
• Know the definitions for one-to-one and onto transformations, and make sure you know
how to determine if a matrix transformation satisfies these properties.
• Note: Make sure that you KNOW how to determine the transformations that are
required to manipulate the basis vectors e1 , e2 ∈ R2 in order to obtain a given matrix
A ∈ R2×2 . Know how to identify the type of transformation, the matrix that yields the
transfomation, and the matrix obtained after performing the transformation. The class
as a whole did poorly on this question, and I can guarantee that you will see one on
your exam!
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6. Other Matrix Properties:
• Section 2.1: Covers matrix-matrix multiplication and matrix transposes.
• Section 2.2: Know how to determine the inverse of a matrix using both Theorem 4 (p.
103) and the algorithm of Example 7 (p. 106). Also, know how to determine the solution
to the matrix equation Ax = b using A−1 , the inverse of the matrix A. Finally, know
how to apply the various properties for the transpose and the inverse of A.
• Note: The inverse of matrix does NOT always exist. Make sure you know how to
determine if a given matrix A has an inverse.
• Section 2.3: Theorem 8 (p. 112) does a fantastic job of summarizing the important
concepts we have covered about the linear independence/dependence of a matrix A and
the properties associate with its corresponding linear transformation T (x) = Ax. Also,
know how to determine the inverse of of a linear transformation T : Rn → Rn , and how
to check if a given transformation S : Rn → Rn is the inverse of T .
• Section 2.4: Know how to perfrom the following properties for partioned matrices:
(a)
(b)
(c)
(d)
Addition and subtraction of partioned matrices
Multiply a partioned matrix by a scalar c ∈ R
Multiply two block matrices
Find the inverse of a partioned matrix using the methods shown in this section.
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