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set of vectors is linearly independent
the equation x1v1 + … + xpvp = 0 has only the trivial solution x1 = … = xp = 0. None of the vectors can be written as a combination of the others
homogenous system
all entries in the answer column are 0. either 1 solution or infinitely many
set of vectors is linearly dependent
there exist constants x1, …, xp, not all zero such that x1v1 + … + xpvp = 0. at least one of the vectors can be written as a linear combination of the others
transformation
a function where the inputs and outputs are vectors
image
output of a linear transformation
preimage
input the output of a transformation corresponds to
elementary vector ei
the vector that has a 1 in the ith position and 0s everywhere else
a transformation Rm→Rn is onto
if each element of b in Rn is mapped to by an element in Rm
a transformation Rm→Rn is one to one
if each element b in Rn is the image of at most one x in Rn
a transformation is linear if
T(x): Rn→Rm is of the form T(x) = Ax for an mxn matrix A
IF T(u+v) = A(u+v) = Au + Av = T(u) + T(v)
IF T(cu) = cT(u)
If T(x) is a linear transformation, then T(0) =
0
A is a diagonal matrix if entries not along the diagonal are ___, and entries on the diagonal are ___
0, not 0
If A is nxn it is
square
matrix A is upper triangular if there is a triangle of zeros at the
bottom
matrix A is lower triangular if there is a triangle of zeros at the
top
an nxn matrix A is invertible (nonsingular)
if there exists an nxn matrix C so that CA = AC = I. C is the inverse of A
If A is nonsingular, A’ is
unique
If A is nonsingular, its determinant is
not 0.
cofactor cij
(-1)^(i+j) * detAij
determinant of an upper or lower triangular matrix
product of the diagonal entries
switching rows does what to the determinant
changes the sign
scaling a row does what to the determinant
scales the determinant the same amount
adding a multiple of another row does what to the determinant
nothing
Points lie on the same plane if the determinant is
0