finals: sections 1.5 - 9.2 Flashcards
(32 cards)
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)
