Week 10 Flashcards
How to show a set (lin span) is in a subspace?
-scalar addition of vectors in set, satisfies subspace
where x = t1(f1) +t2(f2) + t(3f3) where f1,f2 is lin span
To show that a set spans a subspace?
Show that lin span = Subspace
showed through set (lin span) is a subset of the subspace
and the subspace is a subset of the lin span
To show that a subsace spans a set?
use the free parameters from paramtetric equation
Coordinate vector? (V)b
-using the basis to express the vector in a n x 1 vector
How to show a set is a basis?
- det does not equal zero
How to find coordinate vector if not obvious?
-Gauss Jordan Method
What is a basis?y
-a linearly independent set which spans the subspace
How to show basis is linearly independent (function)?
the only solution to the homogeneous equation - c1v1 + c2v2 + … + cnvn =0
is c1=c2=0
How to show basis is spans the subspace (function)?
show in general any a+bx can be written as scalar f1 + scalar f2 , then rearrange so any g can be written when scalar is in terms of a and b
What 3 propretires do an inner product have?
1) linearity on left
2) symmetry
3) positivity
How to prove linearity on left?
<αx + βy, z> = α <x,z> + β<y, z>
How to prove symmetry?
<x,y> = <y,x>
How to prove positityv
<x,x> > 0 and if <x,x>=0 , is if x = zero vector (0,0,0)
l l V l l ? (length/norm)
root (<v,v>)
cos θ?
<u,v> / ll u ll ll v ll
Two vectors orthogonal?
dot product = 0
Norm of one orthogonal base?
1
Gran- Schmidt Method?
u1 = v1 / ll v1 ll
w2 = v2 - <v2,u1>u1
u2 = w2 / ll w2 ll
w3 = v3 - <v3,u1>u1 - <v3,u2>u2
u3 = w3 / ll w3 ll
