Unit 4

Question 1

4.1

what are stretch directions o a LT?

Answer 1

Are the lines along which a linear transormation stretches/contracts or relect

no rotate

Question 2

4.1

How can the amount that linear transormation stretches or contracts in any stretch direction be quantified?

Answer 2

finding a vector v(stretch-direction) lying in the line and solve or landa (the streching actor)

Question 3

4.1

What happens to the overall rotation when a linear transformation has multiple stretch directions?

Answer 3

the more stretch directions the less overall rotation.

Question 4

4.1

what does it mean if the stretch factor is negative?

Answer 4

means a relection

Question 5

4.1

can a linear transormation can have no stretch factors?

Answer 5

no strech factors indicates a significant rotation

Question 6

4.1

what does a stretch factor of 0 means?

Answer 6

it indicates the collapse of a stretch direction

Question 7

4.1.1

How do u find the eigenvectors and eigen values algebraically?

Answer 7

USing this equation (T- λI)v=0 adn then find its determinant that must be equal to 0

v must be a non-zero/ λ=eigenvalue/ v=eigenvector

algebraically talking

Question 8

4.1.1

When the equation (T- λI)v=0 is true?

Answer 8

Only can be true if has a nontrival kernel. This means that the det((T- λI)) must be 0, not invertible.

Question 9

*

How are the basis for an eigenspace found?

Answer 9

they are the same as the eigen vectors

Question 9

4.1.1

what is the difference btw algebraic and geometric multiplicity?

Answer 9

(λ-2)^2=0
algebraic= (λ-2)^2 =>2
geometric= 1 -> λ=2

they not need to be the same

but check

Question 10

4.2

Does λ could have no real solutions?

Answer 10

yes, they gave us magnitud and angel of rotation

Question 11

4.1.1

what are the degrees o freedom?

Answer 11

free variable are gotten equating the eigenvectors, they match the number of basis for the eigenspaces. In other word when we get somehting like 2b-2b=0-> one degree of freedom.
In case we got something like a=b/2. IT is also considered one degree of freedom

Question 12

4.2.1

How do you know the times you need to perform a linear transformation in order to get a real stretch factor?

Answer 12

making the eulerian format equal to e^(pi) you will found how many transormations are need for getting a real stretch factor

Question 13

4.2

Convert the following imaginary number into eulerian(r and θ) format:
2-2i

Answer 13

a+bi
r=raíz(a^2 +b^2)=2raiz(2)
θ=arctan(b/a) =arctan(-1)
re^(iθ)= 2raiz(2)e^(i arctan(-1))

arctan(inf) pi/2

Question 14

4.3

what is a diagonal matrix and what is the main diagonal?

Answer 14

a diagonal matrix is a nxn matrix where all the entries not in the main diagonal are all zero. The main diagonal are the diagonal of entris in a nxn matrix

Question 15

4.3

What notation should we use to indicate that a linear combination is being performed with the stretch basis?

stretch basis = eigenvectors

Answer 15

[a]_uv

Question 16

4.3

how do you translate rom xy system to uv system

Answer 16

vector x=a(u)+b(v)
u=(1)
(1)
v=(0)
(1)

Question 17

4.3

how do u translate from uv system to xy?

Answer 17

using the inverse of uv

uv^-1

Question 18

4.3.1

when a matrix is diagonalizable?

Answer 18

it fiagonalizable i there is a diagonal matrix D
and a invertible C
so that: T =CDC^-1

Question 19

4.3.1

what is D?

Answer 19

D is a diagonal matrix formed by the eigen values of another matrix T

Question 19

4.3.1

what is C?

Answer 19

C is a matrix composed by the eigenvectors

Question 20

why diagonalize?

Answer 20

it makes easy to perform calculations with huge matrices. Specialy or finding matrices to the power of some scalar “a”:
[A]^5=CD^5C^-1

Question 21

4.3.1

if a matrix A is invertible is diagonalizable?

Answer 21

False

Question 22

If matrix A is diagonizable, then it is invertible?

Answer 22

false

Question 23

4.4

what are elipse factors?

Answer 23

the longest and shortest radial lengths found in the image

both are the radius in a circle, in a line longest 1/2, shortest 0

point longest 0, shortest 0

Question 24

4.4

what is a ellipse direction?

Answer 24

any vectors that point in the direction of the longest and shortest radial length

Question 25

4.4

Are stretch and ellipse factor the same?

Answer 25

No,

Question 26

4.4

what represent the total number o ellipse factor?

Answer 26

the dimensions of the codomain

Question 27

4.4

what are the number of non-zero ellipse factor?

Answer 27

total rank of A

Question 28

what is atranspose of a matrix

Answer 28

entris of the matrix reflected across the main diagonal

Question 29

4.4.1

How do you find the ellipse factors algebraically?

Answer 29

by computing the eigen values of AA^t

the eigenvectors are the ellipse directions

Question 30

4.5

When a matrix is symmetrtic, positive definite, positive semidefinite?

Answer 30

is symmetric when it is equal to its own transpose
it is positive definite if its symetric and all the eigenvalues are greater than 0
it is positive semidefinite when it is symmetric and all of the eigenvalues are greater than or equal to 0

Question 31

4.5

What is special about positive definete symmetric matrices?

Answer 31

they are the only matricis whose ellipse factors and stretch factors are the same.
B=B^t
Bv=lampdav
B^2v=lampda^2v

Question 32

4.6

see the last video is too much