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.
What are eigenvalues and eigenvectors for?
They are not for Ax=b they are for du/dt=Au. In the special direction of the eigenvector A acts like a number, the eigenvector, and the problem is one dimensional.
What is a positive definite matrix?
When the eigenvalues are all positive.
What are four special linear combinations?
1) 1v+1w, sum
2) 1v-1w, difference
3) 0v+0w, zero vector
4) cv+0w, vector cv in the direction of v, scalar multiple of v
Add two column vectors
v=[v1;v2] w=[w1;w2] v+w=[v1+w2;v2+w2] or v=(v1,v2) w=(w1,w2) v+w=(v1+w1,v2+w2)
what is a unit vector
a vector that u.u=1
when is v.w=0
then v and w are perpendicular
how to multiply Ax by row
A=(r1,r2,r3)
Ax = [r1.x, r2.x, r3.x ]
how to multiply Ax by column
A=[c1,c2,c3]
Ax= x1c1+x2c2+x3*c3
What is solution to Ax=b
x=(A^1)*b
What is the difference matrix
[ 1,0,0; 1,-1,0; 0,-1,1 ]
What is cyclic difference matrix
[ 1,0,-1; -1,1,0; 0,-1,1 ]
when are u,v,w independent
only 0u+0v+0w=0
when are u,v,w dependent
many combinations of u+v+w give b=0
