Ch. 5 Spanning Sets and Linear Independence

Question 1

What are the conditions of a vector subspace?

Answer 1

Closure under addition
Closure under scalar multiplication
Non-empty

Question 2

What is an affine subspace?

Answer 2

Does not go through the origin and is not a vector subspace

Is the solution set to a non-homogeneous linear system Ax = b (b not equal to 0)

Question 3

What is a spanning set?

Answer 3

The linear span of u1, …. ,uk denoted by span(u1, …. ,uk) is the subset U of R^n consisting of all linear combinations of these vectors:
span(u1, … ,uk) = U = {λ1u1 + …. + λkuk | λ1, … λk in the reals}

Question 4

Definition of linear independence

Answer 4

The vectors u1, … ,uk are linear independent if the only solution to the equation
λ1u1 + … + λkuk = 0
is λ1, …, λk = 0

Question 5

Definition of a basis

Answer 5

Let U be a vector subspace of R^n. If u1, … ,uk are vectors in U that are linearly independent and span U, then S = {u1, … ,uk} is called a basis of U

Question 6

What is the column rank of an m x k matrix A?

Answer 6

The dimension of the span of the columns of A (the column space, which is a subspace of R^n)

The number of linearly independent columns

Column rank = Row rank = Column and Row rank of A^t

Question 7

What is the row rank of an m x k matrix A?

Answer 7

The dimension of the span of the rows of A (the row space of A, which is a subspace of R^k)

The number of linearly independent rows

Row rank = Column rank

Question 8

What is the nullity of an m x k matrix A?

Answer 8

The dimension of the nullspace/kernal of A, which is the solution space to the homogeneous system Ax = 0

Question 9

What is the rank-nullity theorem for A ∈ Mm,n(R) and A^t ∈ Mn,m(R)

Answer 9

n = rk(A) + nullity(A)
m = rk(A^t) + nullity(A^t)

Question 10

dim(U+V) =

Answer 10

dim(U) + dim(V) - dim(U n V)