What does the dot product measure?
how similar two normalized vectors are
How to calculate the magnitude of a vector x? What is its interpretation?
also called euclidean norm
-> length of the vector…
What is the geometrical representation of the dot product?
product of magnitudes of the vectors
times cosine the angle between them
can be used with cos^-1 to find actua angle
-> cos^-1((a*b) / (||a||*||b||))
What is the interpretation of the projection of a vector b onto vector a?
calculating the component of vector b that lies in the direction of a
-> same direction as a
-> magnitude determines how much of the vector lies on a…
=> extend of which b reaches in that direction…
-> extend b downwards to a
-> up to where it crosses…
How to calculate the projection of b onto a?
What is the interpretation of the cross product (a x b)?
a x b is vector that lies orthogonal to a and b
How to calculate the cross product?
How does the cross product change with the angle between a and b?
if a and b orthogonal
-> reaches maximum length
if a and b same or opposite direction
-> length 0
What is the geometric interpretation of the cross product?
calculates the area of parallelogram spanned by a and b
!!!! ||a x b|| !!!
What is the geometrical interpretation of the triple product?
volume of parallelepiped
spammed by vectors a,b,c
-> ||axb|| as grundflächen
projection of c onto (axb) als höhe
verschiebung entsprechend c
How to calculate the triple product (volume)?
How to calculate the kronecker product?
What is the definition of a basis?
set B of vectors is called Basis
in vector space V
if every element in V
may be rewritten in a unique way
as a finite linear combinaiton
of elements of B
What is the linear span?
let S be linear space
let x1,…,xn be n vectors element S
linear span of x1,…,xn
span(x1,…,xn)
=> contains all linear combinations of x1,…,xn
x = a1*x1+ … + an*xn
wher a1,…,an are arbitrary scalars
How is linear independence defined?
set of vectors {v1, v2, …, vk} is linear independent
if x1v1 + x2v2 + … + xnvn = 0
has the only trivial solution x1 = … = xn = 0
What is the goal of the gram-schmidt process?
having a set of vectors (that span some space)
normalize them and orthogonalize them
so that they still span the same space…
How is projection calculated in gram schmidt?
What steps do we take in gram schmidt?
What belongs to euclidean transformatinos?
translation
rotation
(identity)
What belongs to linear transformations?
scaling
reflection
isotropic scaling
What belongs to similarity transformations?
euclidean transformations
What belongs to affine transformatinos?
linear transformations
What belongs to projective transformations?
affine transformations
perspective
How does translation work (including matrix)
How does rotation work (including matrix)?
How does isotropic (uniform) scaling work (with matrix)?
How does scaling work (wiht matrix)?
How does reflection work (with matrix)?
How does shear work (with matrix)?
How to transpose a matrix?
AT_ij = A_ji
How is the rank of a matrix defined?
dimension of vector space
spanned by its columns/rows
corresponds to maximal number of linear indepenedent columns in A
How is the trace of a matrix defined?
Sum over diagonal elements
-> matrix needs to be square
How is the determinant of a matrix defined?
A scalar value that is a function of the entries of a square matrix
sum over positive
product top left to bottom right
plus sum over negative
product top right to bottom left
What is the geometric meaning of a determinant?
absolute value of determinant
2D:
area of parallelogram
3D:
volume of paralellepiped
What is one prominent application of the determinant?
homogeneous system of linear equations has a unique solutoin
if and only if
determinant is non zero
What is the intuitive explanation of determinant being non-zero?
if determinant is 0
-> parallelogram it spans is 0
-> vectors are dependant
-> infinite number of solutions…
(i.e. x-y = 0 when x = y…)
else if independant -> span paralellogram…
What is the kernel/null space of a matrix?
kernel of A is set of vectors {x} satisfying:
all vectors that yield Ax = 0
Is the null-space always empty?
no
-> always non-empty as trivial solution [0,0,0….] satisfies it…
What LGS do we have to sholve when we want to find the kernel of a matri A?
How is the skew-symmetric matrix for a vector a = (a1, a2, a3)T defined?
What can the skew-symmetric matrix be used for?-
to calculate the. cross product
What is the definition of eigenvalues and vectors?
How can one compute eigenvalue and vector?
When does the eigenvalue and vector equation have a nonzero solution v?
determinant of coefficient matrix (A) is zero
-> vectors are linear depentent
How is the characteristic polynomial formed ?
What is the geometrical interpretation of eigenvector and eigenvalues?
eigenvector: changes at most by scalar factor when multiplied with A
eigenvalue: the factor by which the eigenvector is scaled
When is a nxn matrix called invertible?
if there exists nxn square matrix B
such that:
When is a matrix invertible w.r.t. eigenvalues and vectors?
if A can be eigen-decomposed
and none of its eigenvalues are 0
How is the inverse of a matrix given?
Q: nxn matrix
i-th column is eigenvector of A
A(kein A): diagpnal matrix
diagonal elements are corresponding eingenvalues
How is singular value decomposiiton defined?
=
What is a linear map?
nxn matrix represnet linear map T
T: R^n -> R^n
What is the use of SVD?
find linear map T
so shat basis stays orthogonal
after transformation Tv
SVD genauer anschauen…!
What is the QR decomposition?
decomposition of matrix A
into a product A = QR
of an orthogonal matrix Q
and an upper triangular matrix R
When is a matrix Q orthogonal?
columns are orthogonal unit vectors (length 1)
satisfying Q^T = Q^-1
Whe is an upper triangular matrix?
has the form
How can we calculate the QR decomposition? (Q part)
apply Gram Schmidt process to
A = [a1, …, an]
then normalize the resulting vectors to unit vectors
-> Yields Q
How can we calculate the QR decomposition? (R part)
When is a linear system of equations homogeneous?
Right hand side is 0
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