Chapter 1 Intro to Vectors Flashcards
what is the picture of all combinations of cu
the combinations of cu fill a line through [0]
what is the picture of all combinations of cu+dv
the combinations of cu fill a plane through [0]
what is the picture of all combinations of cu+dv+ew
the combinations of cu fill a space through [0]
what happens if w can be described by cu+dv
we do not get the extra dimension
what is true when the dot product is zero
the vectors are perpendicular
how to find the length of a vector
dot product of vector with itself is length squared
find cos(a) for vectors u and v
cos(a) = (u.v)/(|u||v|)
what is angle between [1 0] and [1 1]
cos(a)=1/((1)(Sqrt[2])) therefore a=45d
what is an inner product
another name for dot product, it is inside a matrix
since |cos(a)|<=1 what is true about dot product
|u.v| <= |u||v|
how to compute u.v
u1v1+u2v2
is dot product commutative?
yes
what is a unit vector
a vector with length 1 u.u=1
how to find length of v in matlab
norm(v)
5 basic problems in LA
1) Ax=b -> find x
2) Ax=λx -> find λx
3) Av=σu -> find v,σ,u
4) minimize ||Ax^2||/||x^2||
5) Factor A
How do we know is Ax=b has a solution
Is b in the column space of A? Then yes
What is SVD?
find simplest σuv^T
What is principle component analysis?
find all σuv^T
What is the column space?
The linear combination of all the coumns
Procedure to make C, column space, a basis of A
1) if c0 of A is not 0 put it into C
2)for each Cx of A, if Cx is not a combination of C put it into C
C will have r columns <= n, r is rank of A
Define rank
number of independant rows and columns of A
A=CR R=rref(A)
A->(m x n) -> (m x r)(r x n)
Define symmetric matrix
S=S^T, for all i,j Si,j=Sj,i
Define orthognal matrix
Q^T=Q^-1, qi,j={0 for i!=j, 1 for i=j}
Describe the elimination process
E=series of elimination steps. Start with A, multiply by E, end with U. AE=L and A=LU. Lower triangular L holds forward elimination steps, and U is upper triangular for back substitution.
