Unit 4 Flashcards
4.1
what are stretch directions o a LT?
Are the lines along which a linear transormation stretches/contracts or relect
no rotate
4.1
How can the amount that linear transormation stretches or contracts in any stretch direction be quantified?
finding a vector v(stretch-direction) lying in the line and solve or landa (the streching actor)
4.1
What happens to the overall rotation when a linear transformation has multiple stretch directions?
the more stretch directions the less overall rotation.
4.1
what does it mean if the stretch factor is negative?
means a relection
4.1
can a linear transormation can have no stretch factors?
no strech factors indicates a significant rotation
4.1
what does a stretch factor of 0 means?
it indicates the collapse of a stretch direction
4.1.1
How do u find the eigenvectors and eigen values algebraically?
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
4.1.1
When the equation (T- λI)v=0 is true?
Only can be true if has a nontrival kernel. This means that the det((T- λI)) must be 0, not invertible.
*
How are the basis for an eigenspace found?
they are the same as the eigen vectors
4.1.1
what is the difference btw algebraic and geometric multiplicity?
(λ-2)^2=0
algebraic= (λ-2)^2 =>2
geometric= 1 -> λ=2
they not need to be the same
but check
4.2
Does λ could have no real solutions?
yes, they gave us magnitud and angel of rotation
4.1.1
what are the degrees o freedom?
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
4.2.1
How do you know the times you need to perform a linear transformation in order to get a real stretch factor?
making the eulerian format equal to e^(pi) you will found how many transormations are need for getting a real stretch factor
4.2
Convert the following imaginary number into eulerian(r and θ) format:
2-2i
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
4.3
what is a diagonal matrix and what is the main diagonal?
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
4.3
What notation should we use to indicate that a linear combination is being performed with the stretch basis?
stretch basis = eigenvectors
[a]_uv
4.3
how do you translate rom xy system to uv system
vector x=a(u)+b(v)
u=(1)
(1)
v=(0)
(1)
4.3
how do u translate from uv system to xy?
using the inverse of uv
uv^-1
4.3.1
when a matrix is diagonalizable?
it fiagonalizable i there is a diagonal matrix D
and a invertible C
so that: T =CDC^-1
4.3.1
what is D?
D is a diagonal matrix formed by the eigen values of another matrix T
4.3.1
what is C?
C is a matrix composed by the eigenvectors
why diagonalize?
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
4.3.1
if a matrix A is invertible is diagonalizable?
False
If matrix A is diagonizable, then it is invertible?
false
4.4
what are elipse factors?
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
4.4
what is a ellipse direction?
any vectors that point in the direction of the longest and shortest radial length
4.4
Are stretch and ellipse factor the same?
No,
4.4
what represent the total number o ellipse factor?
the dimensions of the codomain
4.4
what are the number of non-zero ellipse factor?
total rank of A
what is atranspose of a matrix
entris of the matrix reflected across the main diagonal
4.4.1
How do you find the ellipse factors algebraically?
by computing the eigen values of AA^t
the eigenvectors are the ellipse directions
4.5
When a matrix is symmetrtic, positive definite, positive semidefinite?
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
4.5
What is special about positive definete symmetric matrices?
they are the only matricis whose ellipse factors and stretch factors are the same.
B=B^t
Bv=lampdav
B^2v=lampda^2v
4.6
see the last video is too much
