Semester 2

Question 1

What is a vector space?

Answer 1

Let F be a field of scalars. A non-empty set V of vectors, with two operations:
\+: VxV -> V (vector addition)
.: FxV -> V (scalar multiplication)
is an F-vector space if
1. (V, +) is an abelian group.
2. λ(u+v) = λu + λv (λ in F, u,v in V)
3. (λ + μ)v = λv + μv
4. (λμ)v = λ(μv)
5. 1v = v

Question 2

What is a vector subspace?

Answer 2

Let V be an F-vector space. An F-vector subspace of V is a subset U of V that is a vector space under the same operations + and . .

1. U not the empty set.
2. u, v ∈ U implies that u + v ∈ U.
3. u ∈ U implies that λu ∈ U. (λ in F)

Question 3

Row/Column Reduction rules

Answer 3

1. Swap rows/column.
2. Multiply by scalar.
3. Subtract scalar multiple of one from another.

Question 4

Define span (of an F-vector space).

Answer 4

Let V be an F-vector space, S={v1, …, vr} a finite subset of V. The Span of S,
span S = {x1v1 + … + xrvr : x1, …, xr ∈ F}.
The set of all linear combinations of v1, .., vr.

Question 5

Define Reduced Row Echelon Form

Answer 5

1. All zero rows at bottom.
2. Leftmost entry is 1 (pivot/leading).
3. Leading 1 is strictly left of above pivots.
4. Column containing leading 1 has zero in all other entries.

Question 6

What is the nullspace?

Answer 6

Given a field F, and a matrix A∈M(m,n)(F),
Nulll(A):={x∈F^n : Ax = 0}.
Null(A) is an F-vector subspace of F^n.

Question 7

Define a basis.

Answer 7

A span that is linearly independent.

Question 8

What is a linear map?

Answer 8

Let F be a field and V, W be F-vector spaces. A map l: V -> W is F-linear is:
1. l(u+v) = l(u) + l(v).
2. l(λv) = λl(v)
A composition of two linear maps is also linear.

Question 9

Define a linear isomorphism.

Answer 9

A linear map that is also a bijection.