North Carolina Central University
Department of Mathematics and Computer Science
MATH 4410 Linear Algebra I
Instructor Dr. G.Melikian
Name - - - - - - - - - - - - - - - Secton #- - - Problem #1 (30 pts) Determine if the following statements are tr
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
Determinants. Cofactor Expansion
Def. 1. Let
be a matrix. The scalar is called the
DETERMINANT of the matrix and denoted by or . So, we
Ex. 1. Compute the determinant of the given matrix
Def. 2. We define the determinant of an mat
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
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
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
Composition and Invertibility of Linear 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.
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
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
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
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