Vector spaces and linear combinations

Question

Question:

Vector Spaces and Linear Combinations

Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R

Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.

1. Label the following statements as true or false

a) The zero vector is a linear combination of any nonempty set of vectors

b) If S is a subset of a vector space V, then span (S) equals the intersection of all subspaces of V that contain S..

2.In each part, determine whether the given vector is in the span of S.

a) (2,-1), S = {(1,0,2),(-1,1,1)}

b) (-1,2,1), S = {(1,0,2),(-1,1,1)}

c) (-1,1,1,2), S = {(1,0,1,-1),(0,1,1,1)}

d) (2,-11,-3), S = {(1,0,1,-1),(0,1,1,1)}

e) -x³ + 2x² + 3x + 3, A = {x³ + x² + x + 1,x² + 1,x + 1}

3) In F^n, let e denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {e1,e2,...,en} generates F^n.

4)Label the following as true or false statements true or false.

a) If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S
b) Any set combination the zero vector is linearly dependent.
c) Subsets of linearly dependent sets are linearly dependent
d) Subsets od linearly independent sets are linearly independent.

5)In F^n, let e denote the vector whose jth coordinate is 1 and whose other coordinates are O. Prove that {e1, e2, ...., en}is linearly independent.

6) Give an example of 3 linearly dependent vectors in R^3 such that none of the three is a multiply of another.