applied linear algebra exam

How to find determinant of A

a b

c d

ad - bc

How to find inverse of A for 2×2 matrix

1/detA * [ d -b | -c a]

how is a matrix symmetric

A = A^T

what is the format of an inconsistent entry

[0 0 0 … b!=0]

parametric form

p + (t1 * v1) + (t2 * v1) + etc

the three solution types for systems

no solution, a unique solution, infinite

how do you get no solution

inconsistent

how do you get unique solution

no free variable

how do you get infinitely many solutions

at least one free variable

how to find inverse matrix of anything bigger than 2×2 using identity matrix

[A | I ] and find RREF

If A is invertible, then the matrix equation _______ can be solved by computing _______

AX =B, X=A^-1 B

(AB)^T =

B^T * A^T

(A^T)^T =

A

(A+B)^T=

A^T + B^T

(AB)^-1 =

B^-1 A^-1

(A^-1)^-1 =

A

(A^-1)^T =

(A^T)^-1 =

closed under addition

the sum of u and v is in V

commutative property

u+v=v+u

associative property

(u+v) + w = u + (v+w)

additive identity

there is a zero vector in V such that u + 0 =u

additive inverse

for each u in V there is a -u such that u + (-u) = 0

scalar multiplication

scalar multiple of u by c , uc is in V

distributive properties

c(u+v) + cu + cv

(c+d)u = cu + du

property c(du) =

(cd)u

property 1u=

u

3 properties of subspaces

0 vector, closed addition, closed multiplication

In the form [A | b] what is A

the coefficient matrix

what is a homogeneous system

has to equal 0 vector