linear algeb notes

8/10/2019 Linear Algeb notes

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the nullity of T measures how much information (or degrees of freedom) is lost when applyingT .

example: π(x1,...,x5) = (x1, x2, x3, 0, 0)null space dimention 2. exactly the freedom to vary x4 and x5 are lost after applying π.

T : V 7→ W , the rank of T is the dimension of R(T ) – rank(T ) = dim(R(T ))

rank measures how much information (or degree of freedom) is retained by T.

nullity(T ) + rank(T ) = dim(V )

corollary: if dim(V ) = dim(W ), then linear transformation T : V 7→ W is one-to-one iff T isonto.

if T : V 7→ W is linear, and {v1,...,vn} spans V , then {T v1,...,Tvn} spans R(T ).if T : V 7→ W is linear and one-to-one, and {v1,...,vn} is linearly independent, then

{T v1,...,Tvn} is linearly independent.if T : V 7→ W is both one-to-one and onto, and {v1,...,vn} is a basis of V , then {T v1,...,Tvn}

is a basis of W .ex. T : P 3(R) 7→ R4 defined by T (ax3 + bx2 + cx + d) = (a,b,c,d)

so convert basis of P 3(R) to a basis in R4. {1, x , x2, x3} to {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0, )...}

use basis to describe linear transformation in a compact way.

Let V be a vector space with basis {v1,...,vn}. Let W be another vector space and let w1,...,wn

be arbitrary vectors in W . then there exists exactly one linear transformation T : V 7→ W

such that T v j = w j for j = 1 − n.

standard ordered basis for RN : (e1,...,en)

let β = (v1,...,vn) be an ordered basis for V , then ∀

v ∃

! v = a1v1 + ... + anvn. define coordinate

vector of v relative to β as [v]β = [a1

...

an

]

thus we represent any vector as a familar column vector, provided that we supply a basis β .units of measurements are to scalars as bases are to vectors.

let V, W be finite dimensional vector spaces, and let β = (v1,...,vn) and γ = (w1,...,wm) beordered bases for V and W .

take a vector in V , [v]β = [x1

...xn

]

similarly, [T v]γ = [y1...

ym

]

v = x1v1 + ...

T v1 = a11w1 + ... + am1wm

...

1



y1...ym

=

a11 a12 ... a1n

a21 a22 ... a2n

am1 amn

x1

...xn

[T ]γ β is defined to be that matrix, and [T ]γ β = ([T v1]γ [T v2]γ ...[T vn]γ )and recall {v1,...,vn) is denoted as β .

[T v]γ = [T ]γ β[v]β

Eisenstein summation convention

[T ]γ β is the matrix representation of T wrt β and γ . the j th column is just the coordinate vectorof T v j wrt γ .

L(V, W ) ⊂ F (V, W )

S : V 7→ W , T : U 7→ V , then S (T ((u)) is another transformation.

[S + T ]γ

β = [S ]γ

β + [T ]γ

β[cT ]γ β = c[T ]γ βmore importantly,T : X 7→ Y , S : Y 7→ Z

[ST ]γ α = [S ]γ β[T ]βα(after applying T , vector in α basis transfers to vector in β basis, then after S , it transfres to

γ basis)

if A is m ∗ n, then I mA = A and AI n = A

summary:

given a vector space X and an ordered basis α for X , we can write v ∈ V as column vectors [v]α

,given two vector spaces X and Y and ordered bases α and β , we can write linear transformationsT : X 7→ Y as matrices [T ]βα, the action of T is interpreted as:

[T v]β = [T ]βα[v]α

similarly, composition of two linear transformations correspond to matrix multiplication: if S : Y 7→ Z and γ is an ordered basis for Z , then

[ST ]γ α = [S ]γ β[T ]βα (matrix multiplication)

let A be m ∗ n matrix, then define LA : Rn 7→ R

m by: LAx = Ax for x ∈ Rn

matrix multiplication rules translate into linear transformation composition rules.

handwriting for 2 pages?

change of basis, how is [v]β and [v]β0

related?

2



use equation I V v = v, convert this equation into matrices, measuring the domain V using β but range V useingβ 0

[Iv ]β0

β [v]β = [v]β0

recall that [Iv ]β0

β is applying I V to elements of β and writing them in terms of β 0 .

[I V ]β0

β is the change of coordinate matrix from β to β 0. change of coordinate matrices are always squareand invertible.

linear T : V 7→ V. given a basis β of V , we can write a matrix [T ]ββ representing T in the basis β . if we use

another basis β 0 we get a diff erent matrix [T ]β0

β0 , which is related to [T ]ββ.

let Q = [I V ]β0

β , then [T ]β0

β0 = Q[T ]ββQ−1

pf: [T ]β0

β0 = [I V ]β0

β [T ]ββ [I V ]ββ0

two n ∗ n matrices A and B are similar if B = QAQ−1 for some invertible Qn∗n.

recall the rank of a linear transformation T : V 7→ W is the dimension of its range R(T ).

lemma: suppose dim(V ) = dim(W ) = n. then T is invertible iff rank(T ) = n.

pf: if rank(T ) = n, then R(T ) has the same dimenstion as W . hence onto. from dimension theorem we see thatnullity(T ) = 0, so T is 1-1.

lemma: let T : V 7→ W be a linear transformation, S : U 7→ V be an invertible transformation, Q : Q 7→ Z

another invertible. then r(T ) = r(QT ) = r(T S ) = r(QT S )so if you multiply a linear transformation on the left or right by an invertible transformation, then the rank

doesn’t change.

to compute the rank of an arbitrary linear transformation is difficult. the best way is to convert the transformto a matrix, whose rank can be calculated. Let A be a matrix in row-echelon form, then rank(A) = # of non-zerorows

Let Am∗n has rank r. then we can use elementary row and column matrices to place A in the form ( I r 0r∗(n

0(m−r)∗r 0(m−r)∗

let Am∗n with rank r. then we have Bm∗m and C n∗n which are products of elementary matrices (corresponding

to row operations), hence invertible, such that: A = B( I r 0r∗(n−r)

0(m−r)∗r 0(m−r)∗(n−r))C.

this is an example of factorization theorem, which takes a general matrix and factors into simpler pieces.

to mimic the properties of linear transformations, we have for matrices:let Am∗n, Bm∗m invertible, C n∗n invertible, then rank(A) = rank(BA) = rank(AC ) = rank(BAC )

if A is invertible, then I = I t = (AA−1)t = At(A−1)t implying At also invertible, and has the same rank as A

as a consequence of factorization example.

let T

: V 7→ W

be a linear transformation, and β

and γ

be bases for V

and W

. then rank

(T

) = rank

([T

]

γ

β

3

linear algeb notes

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