matrix midterm 2

In terms of composition what does S followed by V mean?

V · S

• cause it means apply transformation to s then apply to v so

s(x) → V(s(x)) which is V dot S

in terms of composition what does S following V mean?

S · V

• cause it means apply transformation to v then apply to s so

v(x) → S(v(x)) which is S dot V

what are varibles?

number of coloumns

List of allowed elementary row operations

• swap rows

• multiply row by a constant

• Ri → Ri + cRj

not allowed

• Ri → cRi + Rj

Column of zeros + free varibles( key reminder)

• column of zeros does not mean the variable = 0

• Just means that the varible never appears in eqs, so nothing restricts it

• In the solution vector, this free variable shows up only in its own position, and all other entries are 0.

• here if x1 is free varible s, only x1=S
  

what does it mean if u have more rows than columns

• more equations than unknowns

• The maximum rank of the matrix is the smaller dimension, so rank has to be less than or equal to number of col

• matrix still can potentially be IND

what to do with zero row?

• most of time ignore

• adds no new info - doesnt count toward rank

homogeneous system def

system of linear eqs where right hand side is all zeros:

Ax=o
(row reduce with answer side all 0)

• 2 possible solutions

1. only trivial solution ( all variables, x1,x2,x3 =0- it’s IND)

2. INF many solutions, if theres free variables

matrix algreba action words!!

Find the span

describe all combinations

Find a basis

remove dependent vectors

Find dimension

count independent vectors

Solve Ax=b

row reduce

Find the null space

solve Ax=0

Find a homogeneous system for a span

convert span → equations

rotation matrix

stretching standard matrix

stretches are in 1 direction

contractions and dilation standard matrix

transforms vectors in all directions by same factor

horizontal shear standard matrix

force applied to rectangle causing it to deform into parallelogram

vertical shear standard matrix

force applied to rectangle causing it to deform into parallelogram

reflection standard matrix

general reflection equation

if p does not lie in the plane, f(p)= a vector that corresponds to opp side of the plane

f(p) and p are connect by a line that is perp to the plane of reflection , they are equal distance from plane

if you are not given point p, just use standard basis vectors to find standard reflection matrix!!

about the eq: subtracting p from the projection on n once removes it, subtracting it again flips it to opp side i think?

Range(L)

Range is every transformation of L onto every possible x.

• all outputs that survive linear mapping, L,

Nullspace(L)

Nullspace of linear mapping , L, are all vectors who get transformed into zero vector by L

• another word for Null space is kernal

• often described by set of parameters

4 fundamental spaces + descriptions

Col space: Range of linear mapping

Null space: all vectors that get crushed to 0 thru linear mapping

Row space: All non zero rows of reduced matrix?

Left null space: Null (A^T) , all vectors that get crushed to 0 thru col space transposed

Dimension count formula ( general)

Dim of span(# of pivots) + num of IND eqs(free cols) = R^n ( pls rearrange as needed!)

3 inverse properties

(sA)^-1 = 1/s A^-1

(AB)^-1 = B^-1 A^-1

(A^T)^-1 = (A^-1)^T

determinant of 2×2 matrix

determinant of 3×3 matrix

Sign pattern follows:

+-+

-+-

+-+….

determinant of larger matrices

• row reduce to upper triangle ( along diagonal ofc)

row reducing operation rules:

• if u swap 2 rows, determinant changes sign

• if u multiple row by k, determinant gets multiplied by k

• if u add multiple of one row to another, nothing changes

once row reduced to upper triangle

• multiply the diagonal

Big theorem ( partly)

If A is invertible the det(A) ≠0

main property of determinants

det(AB)=det(A)det(B)

what does more cols than rows mean?

• usually has INF solutions ( if consistent)

• cannot be lin IND

• null space is non trivial ( there are free cols!)

matrix multiplication transposed rules

(AB)^T = B^T A^T order specific ofc!

what defines if a mapping is linear?

if it is closed under linear combination!

• preserves addition

• preserves scalar multiplication

finding row space

Row reduce matrix

take non zero rows of resulting matrix

these rows form basis of row space

repersent as cols

rank nullity theorem

Rank(A) + dim(Null(A)) = number of coloumns ( not dim of A)

Converting to coordinate notes

• easier to convert smth non standard into the standard basis ( rather than vice versa)

• Nonstandard basis can be exspressed simply by coefficants = standard basis

• create matrix

• then use matrix to convert other coordinates ( most of time need to find inverse matrix converting from standard to non standard)

ORR
- find the coefficiants that equal the coordinate were solving for in terms of the basis we’re converting udner

det rules

• det(A) = det(A^T)

• for A to be invertible det(A)=/=0

• if any rows or cols are all 0, det(A)=0

• det(kA) = kⁿdet(A) where n is the size of the matrix

• If A is triangular (upper or lower) → det = product of diagonal entries

• If A is invertible → det(A⁻¹) = 1/det(A)

• determinate of identity matrix is always 1

determinate row/cols signs pattern

odd rows/cols = +-+-
even rows/cols = -+-+

inverse of gen matrix

5.4 ( det + area of parrelolgram)

det(A) is how a parallelogram scales in area
- if matrix is not invertible, area goes to 0 ( like det)

R³ rotation matrix around x axis ( similar to around z axis)

R³ rotation matrix around y axis

sin flipped

R³ rotation matrix around z axis

standard one

how to rotate aroun multiple axis

multiplying rotation matrices together - order matters ( its a composition of funtions) so think of it like rotate around X axis followed by Z axis which means [Z][X]

det inverse rule

If A is invertible… det(A^-1) = 1/det(A)