Chapter 4.2

Question 1

Subspace

Answer 1

A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V

Question 2

What are the following conditions for W that must hold in order for W to be a subspace of V? (2)

Answer 2

1) If u and v are vectors in W then u + v is a vector in W

2) If k is any scalar and u is a vector in W then ku is a vector in W

Question 3

Every vector space R^(n) contains the following subspaces

Answer 3

{0], subspaces of order n-m where m is an integer in [1,n]

Question 4

Examples of subspaces of Mnn (3) (and one example of a subset that is not a subspace

Answer 4

1) Symmetric matrices
2) Triangular matrices
3) Diagonal matrices

!) Invertible matrices

Question 5

if W1 and W2 are subspaces of V then what can be said about the intersection of their subspaces?

Answer 5

It is also a subspace of V

Question 6

The defintition of a linear combination

Answer 6

If w is a vector in V then w is said to be a linear combination of the vectors v1, v2, … , vr in V if w can be expressed in the form

w=k1v1+k2v2+…+krvr

Where k1, k2, … kr are scalars called coefficiants of the linear combination.

Question 7

If S = {w1, w2, … , wn} is a nonempty set of vectors in a vectorspace V, then? (2)

Answer 7

1) The set W of all possible linear combinations of the vectors in S is a subspace of V
2) The set W in part (1) is the “smallest” subspace of V that contains all of the vectors in s in the sense that any other subspace that contains those vectors contains W.

Question 8

What is the span of S?

Answer 8

The subspace of a space V that is formed from all possible linear combinations of the vectors in a nonempty set S is called the span of S, and we say that the vectors in S span that subspace. If S = {w1, w2, … , wn} then we denote thte span of S by span[w1, w2, … , wn} or span{S}

Question 9

The solution set of a homogeneous linear system Ax=0 in n unknowns is also what?

Answer 9

A subspace of R^(n)

Also called a solution space of the system.

Question 10

If S={w1,w2, … , wr} and S’={v1,v2, … , vk} are nonempty set of vectorspace V, then span{S}=span{S’} iff?

Answer 10

Each vector in S can be expressed as a linear dombination of those in S’ and vice versa