Chapter 3 definitions

Question 1

Define what it means for a set S of vectors to be linearly independent

Answer 1

A set S of vectors in V is said to be linearly independent if the only linear
combination of elements of S with summation θ is the trivial linear combination.

Question 2

Define what it means for a set S of vectors to be linearly dependent

Answer 2

A set S of vectors in V is said to be linearly dependent if there are elements
ai ∈ S, for some n ∈ N, and coefficients αi ∈ F, i = 1, . . . , n, not all zero, so
that the linear combination

Question 3

Define a basis

Answer 3

system {v1,v2, . . . ,vn} in V is a basis for V if and only if it is lin-
early independent and spans V , <v1,v2, . . . ,vn> =V .

Question 4

Define a finite dimensional vector space

Answer 4

A vector space V is finite dimensional if there is a finite subset S ⊆ V
such that <S> =V .</S>

Question 5

Define dimension

Answer 5

Let V be a vector space. If V has a basis consisting of n vectors, we
say that V has dimension n.

Question 6

Define a linear combination

Answer 6

A linear combination of vectors v1,v2, . . . ,vn ∈ V is the expression
a1v1 +a2v2, . . .+anvn with a1,a2, . . . ,an ∈ F.