Chapter 4.4

Question 1

What correlation is there between the system Ax=0 and linear independence?

Answer 1

If Ax=0 only have the trivial solution then the row vectors of A are linearly independent

Question 2

Def: Basis

Answer 2

If V is any vector space and S={v1, v2, … , vn} is a finite set of vectors in V, then S is called a basis for V if the following two conditions hold:

1) S is linearly independent
2) S spans V

Question 3

What is the standard basis for Pn and Mmn

Answer 3

1) S={1, x, x², … , x^(n)}

2) m*n vectors where each vector has one entry set to 1 and the remaining to zero.

Question 4

A vector space that has not finite spanning set?

Answer 4

Any infinite-dimensional space, i.e. R^(∞), F(-∞,∞), C(-∞,∞), C^(m)(-∞,∞), C^(∞)(-∞,∞), P∞

i.e. in a set S={v1, v2, … , vr} of polynomial vectors, they would have a finite number of degrees, say n, and would not vover polynomials of degree n+1.

Question 5

Uniqueness of Basis Representation

Answer 5

If S={v1, v2, … , vn} is a basis for a vector space V, then every vector v in V can be expressed in the form v=c1v1+c2v2+…+cnvn in exactly one way.

Question 6

Def: Coordinates of v

Answer 6

If S={v1, v2, … , vn} is a basis for a vector space V and
v= c1v1 + c2v2 + … + cnvn
is the expression for a vector v in terms of the basis S, then the scalars c1, c2, … , cn are called the coordinates of v relative to the basis S. The vector (c1, c2, … , cn) in R^(n) constructed from these coordinates is called teh coordinate vector V relative to S; it is denoted by
(v)s=(c1, c2, … , cn)

Question 7

A one-to-one correspondence

Answer 7

If (v)s is a constructed vector in a vector space V, thses is a one-to-one correspondence between vectors in V and vectors in R^(n) (through the coordinates of V)