Chapter 4.5

Question 1

Number of vectors in a basis of a finite dimensional vector space?

Answer 1

The same as the number of dimensions.

Question 2

Let V be any finite-dimensional vector space, and let {v1, v2, …, vn} be any basis (2)

Answer 2

1) If a set has more than n vectors, then it is linearly dependent
2) If a set has fewer than n vectors then, it does not span V.

Question 3

Def: Dimension

Answer 3

The dimension of a finite dimensional vector space V is denoted by dim(v) and is defined to be the number of vectors in a basis for V. In addition the zero vector space is defined to have dimension zero.

Question 4

Let S be a nonempty set of vectors in vector space V

adding and subtracting vectors from S) (2

Answer 4

1)If S is a linearly independent set, and if v is a vector in V that is outside of span{S}, then the set SU{v} that results by inserting v into S is still linearly independent.
2)If v is a vector in S that is expressible as a linear combination of other vectors in S, and if S-{v] denotes the set obtained by removing v from S, then S and S-{v} span the same space, that is
span(S) = span(S-{v})

Question 5

Let V be an n-dimensional vector space, and let S be a set in V with exactly n vectors. Then S is a basis for V iff

Answer 5

S is linearly independent

Question 6

Let S be a finite set of vectors in a finite dimensional vector space V (2) (correlation between S spanning V and inserting or subtracting vectors)

Answer 6

1) If S spans V but is not a basis for V, then S can be reduced to a basis for V by removing appropriate vectors from S
2) If S is a linearly independent set that is not already a basis for V, then S can be enlarged to a basis for V by inserting appropriate vectors into S.

Question 7

If W is a subspace of a finite dimensional vector space V, then (3)

Answer 7

1) W is finite dimensional
2) dim(W) <= dim(V)
3) W=V if dim(W)=dim(V)