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
