finals: sections 1.5 - 9.2 Flashcards
when is a system of linear equations considered to be homogeneous?
when it can be written in the form Ax = 0
the homogeneous equation Ax = 0 has a nontrivial solution if and only if _____________
the equation has at least one free variable
what is the trivial solution?
the equation Ax = 0, where x = 0 is the only solution
what is a nontrivial solution?
an x does not equal 0 that satisfies Ax = 0
what is parametric vector form?
when the solution is written as a vector equation, x = su +tv
what is linear independence?
a set {v1, …, vp} is linearly independent if the vector equation x1v1 + … + xpvp = 0 has only the trivial solution
the columns of a matrix A are linearly independent if and only if _______________
the equation Ax = 0 has only the trivial solution
if a set has more vectors than entries in each vector, then the set is ___________
linearly dependent
if a set of vectors contains the zero vector, then the set is _________
linearly dependent
when two vectors are multiples of each other, they are _______________
linearly dependent
why do we care about linear independence?
1) it helps us understand whether a set of vectors provides unique information
2) if the set is linearly dependent, some of the vectors are redundant because they can be written as a combination of other vectors
a transformation or mapping, T, is linear if:
1) T(u+v) = T(u) + T(v), for all vectors u and v in the domain of T
2) T(cu) = cT(u), for all scalars c and all vectors u in the domain of T
what is a transformation (or mapping) T from IR^n to IR^m?
a rule that assigns a vector x in IR^n to a vector T(x) in IR^m
what is the domain of T?
IR^n
what is the codomain of T?
IR^m
what is the image of x?
T(x)
what is the range of T?
the set of all images T(x)
what is a shear transformation?
the transformation from IR^2 –> IR^2 defined by T(x) = Ax
what is a dilation?
given scalar r, the transformation from IR^2 –> IR^2 defined by T(x) = rx
what is a diagonal matrix?
a square, nxn matrix with entries on the diagonal and zeros everywhere else
what are the properties of matrices and matrix multiplication?
1) A(BC) = (AB)C
2) A(B + C) + AB + AC
3) (B + C)A + BA + CA
what is the transpose of a matrix?
given an nxm matrix A, the transpose of A, denoted A^T, is the mxn matrix whose columns are formed from the corresponding rows of A
what are the properties of transposing?
1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) (rA^T) = r(A^T), r scalar
4) (AB)^T + B^T * A^T
what is the geometric approach good for?
lower dimensional problems (3D or less)
what is the canonical linear programming problem?
1) maximize c^Tx = f(x) subject to:
2) the constraints of Ax <= b
3) and x > 0
what is an optimal vector?
a vector that satisfies 1, 2, and 3
what is a feasible solution?
a vector that satisfies 2 and 3
what is an objective function?
what you are optimizing
what are the two things that can go wrong in a linear programming problem?
1) infeasible
2) unbounded
what is infeasible?
the constraint inequalities are inconsistent and the set of all feasible solutions F, is an empty set
what is unbounded?
there are arbitrarily large values in F, then a maximum DNE
