midterm 1 Flashcards
when are two systems of linear equations equivalent?
when their solution sets are the same
a system of linear equations has:
1) no solution
2) exactly one solution
3) infinitely many solutions
what are the two fundamental questions about a linear system?
1) is the system consistent?
2) if a solution exists, is it the only one?
what are the conditions for a matrix to be in echelon form?
1) all nonzero rows are above any rows of zeros
2) each leading entry of a row is a column to the right of the leading entry in a row above it
3) all entries in a column below a leading entry are zero
what are the conditions for a matrix to be in reduced row echelon form?
it satisfies all conditions to be in echelon form and:
1) the leading entry in each nonzero row is one
2) each leading one is the only nonzero entry in the column
is the echelon form of a matrix unique?
no
is the reduced echelon form of a matrix unique?
yes
pivot position
the number and spot in the original matrix where a leading one is in the reduced echelon form of the same matrix
pivot column
a column of a matrix that contains a pivot position
when is a linear system consistent?
when it has one solution or infinitely many solutions
when is a linear system inconsistent?
when it has no solution
column vector
a matrix with only one column
row vector
a matrix with only one row
scalar multiple
cu, where u is a vector and c is a scalar
u + v = ______
v + u
(u + v) + w = ______
u + (v + w)
u + 0 = ______
0 + u = u
u + (-u) = _____
-u + u = 0
c(u +v) = _____
cu + cv, where c is a scalar and u, v are vectors
(c + d)u = ______
cu + du, where c, d are scalars and u is a vector
c(du) = _____
(cd)u, where c, d are scalars and u is a vector
1u = ____
u
Span{v1, …, vp} is
the collection of all vectors that can be written in the form c1v1 + c2v2 + …. cpvp
asking whether vector b is in Span{v1, …, vp} is the same as asking _________
if x1v1 + x2v2 + … + xpvp = b has a solution
if matrix A has a pivot position in every row…
the columns of A span R^n
a system of linear equations is homogeneous if:
it can be written in the form Ax = 0
what is the trivial solution?
the vector x = 0 that makes Ax = 0
what is a nontrivial solution?
when vector x does not equal zero and
Ax = 0
parametric vector form
x = su + tv
two matrices are row equivalent if:
there exists a sequence of elementary row operations that transforms one matrix into the other
a system of vectors is linearly independent if _________
the linear combination of the vectors x1v1 + x2v2 + … + xpvp = 0 has only the trivial solution
a system of vectors is linearly dependent if _________
if there exists weights such that c1v1 + c2v2 + … + cpvp = 0, where at least 1 weight is nonzero
when two vectors are multiples of each other, they are __________
linearly dependent
if the set of vectors contains the zero vector, then the set is _______
linearly dependent
if the set contains more vectors than there are entries in each vector, then the set is _______
linearly dependent
a transformation T from IR^n to IR^m is ______
a rule that assigns a vector x in IR^n to a vector T(x) in IR^m
domain of T
IR^n (original)
codomain of T
IR^m (new)
T(x)
the image of x
range of T
the set of all images T(x)
a transformation T is linear if:
1) T(u + v) = T(u) + T(v) for all u, v in the domain of T
2) T(cu) = cT(u) for all scalars c and all u in the domain of T
first part of transformation T being linear
T(u + v) = T(u) + T(v) for all u, v in the domain of T
second part of transformation T being linear
T(cu) = cT(u) for all scalars c and all u in the domain of T
a transformation T from IR^n to IR^m is ______
a rule that assigns a vector x in IR^n to a vector T(x) in IR^m
in the equation Ax = b, what is A and what does A do?
A is a matrix, and it acts on or transforms vector x to produce a new vector b
what does a sheer transformation look like?
it stretches the area of the matrix (the sheep is being pulled)
dilation / contraction
given a scalar r, define T. IR^2 –> IR^2 by T(x) = rx (multiplying the values by a constant doesn’t change anything but the length of the arrow)
dilation criteria
r > 1
contraction criteria
0 <= r <= 1
A(BC) =
(AB)C
A(B + C) =
AB + AC
(B + C)A =
BA + CA
(A^T)^T =
A
(A + B)^T =
A^T + B^T
(rA^T) =
(A^T), r scalar
(AB)^T =
B^T * A^T
two matrices are equal if:
1) they are the same size
2) their columns are equal (all entries are equal)
IA =
A
A^3 =
AAA
canonical linear programming problem
maximize f(x) = c^T * x subject to the constraints Ax <=b and x>= 0
feasible solution
any vector x that satisfies the constraints Ax <= b and x>= 0
feasible set
the set, F, of all feasible solutions
slack variable
a nonnegative variable that is added to the smaller side of an inequality to convert it to an equality
can a slack variable be negative?
NO
when is F an empty set?
when the problem is infeasible
when are there arbitrarily large values in F?
when the problem is unbounded
unbounded
no maximum
infeasible
no solution
