Poole

Question 1

solution of system of linear equations

Answer 1

a vector that is a solution to all the equations in the system

Question 2

consistent

Answer 2

at least one solution

Question 3

[matrix][column vector]=

Answer 3

[column vector]

Question 4

augmented matrix

Answer 4

[A|b]

Question 5

row echelon form if and only if

Answer 5

any all zero rows at bottom
in each non-zero row, leading entry is to the left of any leading entries below it

Question 6

reduced row echelon form if additionally

Answer 6

leading entry in each non-zero row is 1
each column containing a leading 1 has 0s everywhere else

Question 7

elementary row operations

Answer 7

interchange two rows
multiply a row by a constant
add a multiple of a row to another

Question 8

gaussian elimination

Answer 8

form augmented matrix
reduce to reduced row echelon form using EROs
if consistent, use back substitution to solve

Question 9

homogeneous

Answer 9

if and only if the constant term in each equation is zero

solution will always be
[0,0,0]

Question 10

span

Answer 10

set of all possible linear combinations of vectors in S

Question 11

system with augmented matrix [A|b] is consistent if and only if

Answer 11

b is linear combination of columns of A

i.e. b is in the span (s)

Question 12

spanning set

Answer 12

let s={v1,..,vk} be set of vectors in R^n

set is spanning set for R^n if span(s)=R^n

Question 13

if A has n rows, then columns are

Answer 13

vectors in R^n

Question 14

if S is spanning set for R^n then system is

Answer 14

consistent no matter what b is

Question 15

system is consistent for every b in R^n if

Answer 15

S is a spanning set

Question 16

not a unique solution

Answer 16

redundancy

Question 17

linear independence

Answer 17

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

Question 18

linear dependence

Answer 18

if and only if at least one vector can be expressed as a linear combination of the others

Question 19

if m>n then any set of m vectors in R^n is

Answer 19

linearly dependent

Question 20

In

Answer 20

identity matrix

eg:
[10]
[01]

Question 21

matrix multiplication

Answer 21

has to be admissible

i.e. mxn nxp
n must be same

Question 22

matrix powers

Answer 22

A^0=identity
A^2=AA
A^k=AA…A (k times)

Question 23

transpose

Answer 23

mxn becomes nxm
A^T
“rows become columns”

Question 24

symmetric matrices

Answer 24

if A^T=A

Question 25

matrix multiplication properties

Answer 25

A(BC)=(AB)C K(AB)=(KA)B=A(KB) ImA=A=AIn if A is mxn

Question 26

transpose properties

Answer 26

(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

Question 27

if A is a square matrix then A+A^T is

Answer 27

symmetric

Question 28

or any matrix A, AA^T and A^TA are

Answer 28

symmetric

Question 29

trace of a matrix

Answer 29

sum of all the elements in the leading diagonal

Question 30

A^-1

Answer 30

inverse AA^-1 = In = A^-1A *inverse is unique*

Question 31

Ax=b implies

Answer 31

x=A^-1b

Question 32

invertible if

Answer 32

detA does not equal zero

Question 33

A^-1=

Answer 33

1/ad-bc [d -b / -c a]

Question 34

properties of inverse

Answer 34

(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

Question 35

If A is invertible and n greater than or =0, A^-n=

Answer 35

(A^-1)^n=(A^n)^-1