Ch. 6 Real Vector Spaces

Question 1

How to check if a set of matrices or polynomials are linearly independent

Answer 1

Set λ1M1 + λ2M2 + λ3M3 = 0

If the only solution is λ1 = λ2 = λ3 = 0 then they are linearly independent

Question 2

Conditions for u1, … ,uk forming a basis of V

Answer 2

One cannot add another vector in V and still have a linearly independent set
One cannot remove any of the ui and still have a spanning set of V
Any vector u in V can be written uniquely as
λ1u1 + λ2u2 + … + λkuk

Question 3

How to check if a set of vectors is linearly independent

Answer 3

Put them into a matrix and do EROs

If determinant is not equal to zero, they are linearly independent

Question 4

How to check if a set of vectors spans e.g. R^3

Answer 4

put the vectors in an augmented matrix, with x1, x2, x3 on the right
Do EROs to get it in upper triangular form
If there is not a row of zeros, the vectors span

Question 5

What does it mean to say U is a vector subspace of V?

Answer 5

Closure under addition, closure under scalar multiplication

Question 6

What does it mean to say U is an affine subspace of V?

Answer 6

If there exists an x ϵ V and W c V a vector subspace s.t U = x + W