North Carolina Central University
Department of Mathematics and Computer Science
Exam #1
MATH 4410 Linear Algebra I
Instructor Dr. G.Melikian
Name - - - - - - - - - - - - - - - Secton #- - - Problem #1 (30 pts) Determine if the following statements are tr
Section 1.2
Linear Combinations, Matrix-Vector Products,
and Special Matrices
Def.1. A linear combination of vectors u , u ,., u is a sum of
scalar multiples of these vectors, that is, a vector of the form
c , c ,., c are scalars. These scalars are
c u c
Section 3.1
Determinants. Cofactor Expansion
Def. 1. Let
be a matrix. The scalar is called the
DETERMINANT of the matrix and denoted by or . So, we
have
.
(1)
Ex. 1. Compute the determinant of the given matrix
.
Def. 2. We define the determinant of an mat
Section 3.2
Properties of Determinants
We need to know how a determinant will behave if we perform a
linear operation with its rows. Row linear operations will be
used to compute any determinant without cofactor
expansion. We need to find some connection
Section 1.3
Systems of Linear Equations
Def. 1. A system of linear equations is a set of m linear
equations in the same n variables, where m and n are positive
integer numbers. Such a system has the form
a11 x1 a12 x 2 . a1n x n b1
a 21 x1 a 22 x 2 . a 2
Section 1.7
Linear Dependence and Linear Independence
Def. 1. A set of vectors u , u ,., u in the space R is called
linearly dependent if there exist scalars c , c , , c , not all
c u c u . c u 0 . We say also that the
zero, such that
vectors u , u , , u
Section 2.7
Composition and Invertibility of Linear Transformations
Special Transformations
We begin this Section 2.7 with the next useful result.
Proposition 1. The range of a linear transformation
equals the span of the columns of its standard matrix.
P
Section 2.4
The Inverse of a Matrix
We start to answer the questions: (a) when a matrix has an inverse?
(b) how can we find the inverse of a matrix? Let me remind you the
definition of the inverse of a matrix first (see on the board).
Theorem 2.5. Let A b
Section 2.3
Invertibility and Elementary Matrices
Def. 1. An n n matrix A is called invertible if there is an
n n matrix B such that A B I , B A I . At this case matrix B
is called the inverse of A and denoted by A .
Ex. 1. Determine whether B A for the g
Section 2.6
Linear Transformations and Matrices
Def. 1. Let S1 and S 2 be subsets of R n and R m , respectively. A
function f from S1 into S 2 , written f : S1 S 2 , is a rule that
assigns to each vector x S1 a unique vector f ( x) S 2 . The vector
f (x)
Section 4.1
Subspaces
Def. 1. A set W of vectors in R n is called a subspace of R n if
it has the following three properties:
(a) The zero vector 0 belongs to W .
(b) Whenever vectors u and v belong to W , then u v belongs
to W .
(c) Whenever vector u bel