LINEAR ALGEBRA FINAL EXAM REVIEW SET QUESTIONS WITH COMPLETE SOLUTIONS

LINEAR ALGEBRA FINAL EXAM REVIEW
SET QUESTIONS WITH COMPLETE
SOLUTIONS
span - denotes a set of all linear combinations of a set of vectors

homogenous - a system of linear equations is ____ if it can be written in the form Ax=0
for an mxn matrix A

trivial solution - the solution x=0 is called the ____

nontrivial solution - a nonzero vector x that satisfies Ax=0 is called a ____, the
homogenous equation Ax=0 has a ____ if and only if the equation has at least one free
variable

linearly independent - a set of vectors is said to be ____ if the vector equation
x1v1+x2v2+...+xpvp=0 has only one trivial solution

linearly dependent - a set of vectors is said to be ____ if the vector equation
x1v1+x2v2+...+xpvp=0 has more than just the trivial solution

linear transformation - a _____ exists if T(u+v)=T(u)+T(v) and T(cu)=cT(u)

identity matrix - the nxn _____ has 1's on the main left to right diagonal and 0's
everywhere else. the ith column is denoted ei

onto - the mapping T:R^n to R^m is said to be _____ if each b in R^m is the image of at
least one x in R^n, T(x)=b has at least one solution, pivot position in every row

one to one - the mapping T:R^n to R^m is said to be _____ if each b in R^m is the
image of at most one x in R^n, T(x)=b has at most one solution, pivot position in every
column

equal - two matrices are ____ if they have the same size and each of their
corresponding columns are equal

scalar multiple - if r is a scalar and A is a matrix, the the _____ rA is the matrix whose
columns are r times the corresponding columns of A

transpose - if A is an mxn matrix, the _____ of A is the nxm matrix, denoted by A^T,
whose columns are formed from the corresponding rows of A

, invertible, inverse - an mxn matrix A is said to be _____ if there is an nxn matrix C
satisfying CA=AC=In where In is the nxn identity matrix. we call C the ____ of A

singular matrix - a matrix which is not invertible is called a ____

nonsingular matrix - an invertible matrix is called a ____

determinant - the quantity ad-bc is called the ____ of A, and we write detA=ad-bc

elementary matrix - a _____ is one that is obtained by performing a single elementary
row operation on an identity matrix, they are invertible

invertible matrix theorem - for a square nxn matrix A the following statements are
equivalent:
a. A is an invertible matrix
b. A is row equivalent to In (the identity matrix)
c. A has n pivot positions
d. the equation Ax=0 has only the trivial solution
e. the columns of A form a linearly independent set
f. the linear transformation x to Ax is one to one
g. the equation Ax=b has at least one solution for each b in R^n
h. the columns of A span R^n
i. the linear transformation x to Ax maps R^n onto R^n
j. there is an nxn matrix C such that CA=In
k. there is an nxn matrix D such the AD=In
l. A^T is an invertible matrix

invertible - a linear transformation T:R^n to R^n is said to be _____ if there exists a
function S:R^n to R^n such that S(T(x))=x for all x in R^n and T(S(x))=x for all x in R^n

inverse - if it exists, the transformation S is called the ___ of T, and we denote it by
S=T^-1

cofactor expansion - the determinant of an nxn matrix A can be computed by a _____
across any row or down any column

upper triangular matrix - an ____ is one whose entries below the main diagonal are all
zero

lower triangular matrix - an ___ is one whose entries above the main diagonal are all
zero

detA - if a multiple of one row of A is added to another row to produce B, the
detB=_____

-detA - if two rows of A are interchanged to produce B, then detB=_____