7.1 - Linear Independence

Question 1

Def: Linear independence.

Answer 1

Is S = { a1, … , an} spans Rn
S is linearly indpendent for any linear combo of x1a1 + …. + xnan = 0

Question 2

Theory: Steps to prove linear independence of a two vectors?

Answer 2

1. Set up
   - let x1a1 + x2a2 = 0
2. Prove x1 and x2 are unique
   - Create homogenous system and solve for x1 and x2
   - Rank = # of variables means no free variables, so unique soln

Question 3

Def: What is nullity? what is rank + nullity?

Answer 3

Nullity is # of parameters.
rank + nullity = num of col of num of variables

Question 4

Theory: What makes two vectors are linearly independent?

Answer 4

if both vectors are not scalar multiples can cannot be written as liinear combos of each other and are therefore not parallel.

Question 5

Theory: can three vectors in a plane be linearly independent?

Answer 5

No. Any vector in the plane can be written as the linear combination of two other vectors in the plane.

Question 6

Theory: How many solutions can a homogenous system, Ax = 0, have?

Answer 6

One of infinite. the zero vector is always a solution.

Question 7

Theory: 3 properties of a linearly independent set of vectors

Answer 7

S = { a1 … an} spans Rn
1. (a1 … an)x = Ax = 0 has only zero soln so ket(A) = {0}
2. A:Rk to Rn is 1-to-1
3. r(A) - num of variables = nullity

Question 8

Theory: See if three vectors in r3 are linearly independent steps?

Answer 8

1) system Ax = 0
2) Solve system
3) Look for any free variables, is so, then linearly dependent

Question 9

Def: bases

Answer 9

If V is a subspace of Rn
S = {a1 … an} spans V and S is lin ind
S is a bases of V

Question 10

Def: Dimension of span{a1 … an}.

Answer 10

m. If vector space has basis of n elements, the vector space has dimension m
span{a1 … an} has a basis of n elements. If {a1 … ak} is linear indecently is a = k.

Question 11

Theory: Between basis and dim, which are unqiue?

Answer 11

Basis is not unique.
Dim is.

Question 12

Def: What is meant by image? Range?

Answer 12

If a linear transformation, L, that maps RV to RW, vectors in V are transformed into vectors in W. A subspace of V, S, are transformed into vectors in W. The set of vector that we can get from transforming S, is called the image, L(S).

The image of the entire subspace of V is called the range, L(V).

Question 13

Def: What is meant by kernel?

Answer 13

Ker(L) is the set of vectors in vector space V, that when transformed using linear transformation L, will give the zero vector is subspace W.