Chapter 6 Flashcards
Theorem 1
Let u,v, and w be vectors in Rn and let c be a scalar. Then:
a) u.v = v.u
b) (u+v).w=(u.w+v.w)
c) (cu).v = c(u.v)=u.(cv)
d) u.u>= 0 and u.u = 0 if and only if u = 0
Length (Norm)
Nonnegative scalar ||v|| defined by sqrt(v1^2+v2^2+…+vn^2)
unit vector formula
||u||= 1/||v||*v
Orthogonal
Two vectors u and v in Rn are orthogonal if u.v =0
Theorem 2
Two vectors u and v are orthogonal if and only if ||u+v||^2 = ||u||^2+||v||^2
Orthogonal complement
A vector x is in W^ if and only if x is orthogonal to every vector in a set that spans W
W is a subspace of Rn
Theorem 3
Let A be an mxn matrix. the orthogonal complement of the column space of A is the null space of A^T
Angle between two vectors
cosO = u.v/(||u||*||v||)
Theorem 4
If S = {u1,…,up} in Rn is an orthogonal set of nonzero vectors in Rn, then S is linearly independent and hence is a basis for subspace spanned by S
Theorem 5
Let {u1,…,up} be an orthogonal basis for a subspace W of Rn. For each y in W, the weights in the linear combination y = c1u1+…+cpup
are given by ck = y.uj/uj.uj
Orthonormal
A set {u1,…up} is an orthonormal set if it is an orthogonal set of unit vectors.
Theorem 7
Let U be an mxn matrix with orthonormal columns and let x and y be in Rn:
a||Ux|| = ||x||
(Ux).(Uy) = x.y
c) (Ux).(Uy) = 0 if and only if x.y = 0
Theorem 8
Let W be a subspace of Rn. Then each y in RN can be written uniquely in the form y = ^y + z where ^y is in W and Z is in W`
Theorem 9
Let W be a of Rn, let y be any vector in RN, and let ^y be the orthogonal projection of y onto . Then ^y is the closest point in W to y, in the sense that
||y-^y|| <||y-v||
Theorem 10
If {u1,…,up} is an orthonormal basis for a subspace W of Rn then
projwY = (y.u1)u1 + (y.u2)u2+…+(y.up)up
