Exam 3 Flashcards
How do you calculate the angle between 2 vectors?
cos θ = <u> / llull llvll</u>
How do you calculate the projection of u onto v
Projection u = (<u>v) /</u>
v</u>
How do you calculate the distance between u and v?
d(u,v) = llu - vll
Inner Product Properties:
Axiom 3
Axiom 3:
<u> = = c<u></u></u>
Inner Product Properties:
Axiom 4
Axiom 4:
<u> >= 0</u>
<u> = 0 if and only if u = 0</u></u>
Linear transformation?
Show:
preserves addition
preserves scalar multiplication
T(cv) = cT(v)
How do you find ker(T)?
Let x = (x1,…,xn) ε ker(T), then
T(x)=0
…
Solve matrix
Resulting vectors span ker(T)
What is ||u||^2?
<u></u>
How do you calculate ||u||?
Calculate ||u||^2 = <u></u>
Take the square root of the answer</u>
How do you find a basis for range(T)?
A basis for the range(T) is the column space of A
Find REF of A. Leading 1s in REF, corresponding column vectors in A form C(A)
How do you express a vector,v, as a l.c. of vectors in an orthonormal set {u1,…,un}?
v = u1 + … + un
How do you find the transformation of a vector given preimages and transformations of other vectors?
v = c1u1 + … + cnun
T(v) = c1T(u1) + … + cnT(un)
How do you find the standard matrix of T?
T(x1,…,xn)^t = [matrix, lined up properly]
= [coefficients][x’s]
Coefficient matrix is A sub t (std matrix for T)
Euclidean inner-product
Regular dot product
What do you write when you have found eigenvectors of A corresponding to lamda = 3
Solutions to (3I - A)x=0 are x = s(..,..,..)^t + t(..,..,..)^t
Eigenvectors of A corr. to lamda = 3 are nontrivial l.c.’s of (..,..,..) , (..,..,..).
If it is a single eigenvector:
Eigenvectors of A corr. to lamda = 3 are nontrivial scalar multiples of (..,..,..).
