# Math 544 Exam 1 Information

## Topic List (not necessarily comprehensive):

**You will need to know how to define vocabulary words/phrases defined in
class.**

1.1: Matrix representation of a linear system: coefficient matrix, augmented
matrix, elementary

row operations, row equivalence.

1.2: Solving linear systems via **Gauss-Jordan elimination:** echelon and
reduced echelon

forms of a matrix, identifying dependent and independent variables, recognizing
when a

system is consistent/inconsistent.

1.3: Relationship between the number of nonzero rows, the number of leading
1’s, and the

number of columns in an augmented matrix in **reduced echelon form.**
Homogeneous linear

systems. The number of possible solutions to (for example)

1. a general linear system.

2. an m*n system with m < n.

3. a homogeneous system.

1.5: Matrix operations: addition, multiplication, multiplication by scalars,
dot product in R^{n}.

1.6: Properties of matrix addition, multiplication, and multiplication by
scalars. The matrix

transpose and its properties. What is the transpose of a product? What is a
symmetric matrix?

Also vector norm (length) in terms of the dot product.

1.7: Linear combinations, linear dependence/independence: determinination of
whether

a given set of vectors is linearly dependent/independent. Non-singular matrices
(remember,

only square matrices can be singular or non-singular!); conditions equivalent to
nonsingularity

of

1. Ax = θ has only the trivial solution x = θ

2. columns of A are linearly independent

3.
Ax = b has a unique solution.

4. A is invertible.

5. A is row equivalent to the identity, I_{n}.

1.9: Matrix inverses: existence of inverses (see above, e.g., A is invertible
<-> A is nonsingular),

using inverses to solve systems, computing inverses by row reduction, formula
for

inverse of 2*2 matrix, algebraic properties of inverses (e.g., what is the
inverse of a product

of two matrices?) Remember, only square matrices can be invertible.

**Notes:**

•Only s**ets of vectors** can be **linearly dependent/independent**. It does not make
sense to

speak of matrices or systems of equations being linearly dependent/independent.

•Only **systems of linear equations** can be **consistent/inconsistent**. It does not
make sense

to speak of matrices or sets of vectors as consistent/inconsistent.

•Only **squares matrices** can be **singular/non-singular**. It does not make sense
to speak of

a system of equations or set of vectors as being singular/non-singular. The same
applies for

**invertibility.**