Chapter 2 - Vector spaces

Question 1

Definition 2.2 - Subspaces

Answer 1

A non-empty subset W of a vector space (V,+, . ) over a field F(bar) is said to be a subspace if it is closed under + and .
i.e.
1) v ∈ W, w ∈ W => v+w ∈ W
2) α ∈ F(bar), w ∈ W => αw ∈ W

Question 2

Proposition 2.3 (iii) - Sum of subspaces

Answer 2

Let X and Y be subspaces of a vector space V.
Then X+Y := {x+y : x ∈ X and y ∈ Y}

Question 3

Definition 2.4 - Direct sum

Answer 3

Let X and Y be subspaces of a vector space V. X+Y is the direct sum (denoted X ⊕ Y) if the union of X+Y = {0}.

Question 4

Definition 3.1 - Spanning set and finitie/infinite dimensional subspaces

Answer 4

Let V be a vector space and let S be a subset of V.
- The SPAN of S (denoted <S>), is the set of all linear combinations of vectors in S.
- We call S a spanning set for V if <S> = V. (S spans V)
- The vector space V is said to be finite - dimensional if V = <Ŧ> for some finite set Ŧ , and infinite - dimensional otherwise.</Ŧ></S></S>

Question 5

Definition 3.2 - Linear Independence and Dependence

Answer 5

A subset L of a vector space V is said to be linearly independent if, whenever v_1,v_2,…,v_n are distinct vectors in L and one has that αv_1 + αv_2 + … + αv_n = 0 => α_1 = α_2 = … = α_n = 0 and is said to be linearly dependent otherwise.

Question 6

Definition 3.3 - Basis

Answer 6

A linearly independent spanning set for a vector space V is called a BASIS of V.

Question 7

Definition 3.9 - Dimension of a vector space

Answer 7

The dimension of a finite-dimensional vector space V (denoted dim V), is the (common) number of vectors in any of its bases.

Question 8

Proposition 3.10

Answer 8

A vector space V is infinite - dimensional if and only if for any n ∈ N(bar) there exists a set of n linearly independent vectors in V.

Question 9

Proposition 3.12

Answer 9

Let V be a finite-dimensional vector space and let X be a subspace of V. Then, there exists a subspace Y of V such that V = X ⊕ Y. Y is called a linear complement of X in V. (Every subspace of a vector space is complemented.

Question 10

Theorem 4.5

Answer 10

A ∈ M_m,n(Fbar) , b ∈ M_m,1(Fbar)
i) Kronecker - capelli : Ax=b is consistent iff rank(A) = rank(A|b)
ii) If Ax=b is consistent then it has a unique solution if rank(A) = n