MAT205 Exam 2

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