Poole Flashcards
solution of system of linear equations
a vector that is a solution to all the equations in the system
consistent
at least one solution
[matrix][column vector]=
[column vector]
augmented matrix
[A|b]
row echelon form if and only if
any all zero rows at bottom
in each non-zero row, leading entry is to the left of any leading entries below it
reduced row echelon form if additionally
leading entry in each non-zero row is 1
each column containing a leading 1 has 0s everywhere else
elementary row operations
interchange two rows
multiply a row by a constant
add a multiple of a row to another
gaussian elimination
form augmented matrix
reduce to reduced row echelon form using EROs
if consistent, use back substitution to solve
homogeneous
if and only if the constant term in each equation is zero
solution will always be
[0,0,0]
span
set of all possible linear combinations of vectors in S
system with augmented matrix [A|b] is consistent if and only if
b is linear combination of columns of A
i.e. b is in the span (s)
spanning set
let s={v1,..,vk} be set of vectors in R^n
set is spanning set for R^n if span(s)=R^n
if A has n rows, then columns are
vectors in R^n
if S is spanning set for R^n then system is
consistent no matter what b is
system is consistent for every b in R^n if
S is a spanning set
not a unique solution
redundancy
linear independence
if and only if the only solution to
c1v1+…+ckvk=0 is
c1=c2…=ck=0
i.e. cam you add up to make zero without multiplying by zero
linear dependence
if and only if at least one vector can be expressed as a linear combination of the others
if m>n then any set of m vectors in R^n is
linearly dependent
In
identity matrix
eg:
[10]
[01]
matrix multiplication
has to be admissible
i.e. mxn nxp
n must be same
matrix powers
A^0=identity
A^2=AA
A^k=AA…A (k times)
transpose
mxn becomes nxm
A^T
“rows become columns”
symmetric matrices
if A^T=A
matrix multiplication properties
A(BC)=(AB)C
K(AB)=(KA)B=A(KB)
ImA=A=AIn if A is mxn
transpose properties
(A^T)^T=A
(A+B)^T=A^T+B^T
(KA)^T=K(A^T)
(AB)^T=B^TA^T
(A^M)^T=(A^T)^M for integers such that m is greater than or equal to 0
if A is a square matrix then A+A^T is
symmetric
or any matrix A, AA^T and A^TA are
symmetric
trace of a matrix
sum of all the elements in the leading diagonal
A^-1
inverse
AA^-1 = In = A^-1A
inverse is unique
Ax=b implies
x=A^-1b
invertible if
detA does not equal zero
A^-1=
1/ad-bc [d -b / -c a]
properties of inverse
(A^-1)^-1=A
If c does not =0 then (cA)^-1=1/cA^-1
if A and B same size (AB)^-1=B^-1A^-1
(A^T)^-1=(A^-1)^T
If A is invertible and n greater than or =0, A^-n=
(A^-1)^n=(A^n)^-1
