Chapter 4.3

Question 1

The definition of [The Trivial Solution], [A Linearly Dependent Set] and [A Linearly Independent Set]

Answer 1

If S={v1, v2, … , vr} is a nonempty set of vectors in a vector space V, then the vector equation:
k1v1+k2v2+…+krvr=0
Has at least one solution, namely,
k1 = 0, k2 = 0, … , kr = 0
We call this the trivial solution, if this is the only solution then S is said to be linearly a independent set. If there are solutions in addition to the trivial solution, then S is said to be a linearly dependent set.

Question 2

A set S with two or more vectors is (2)

Answer 2

1) Linearly dependent iff at least one of the vectors is expressable as a linear combination of the other vectors in S.
2) Linearly independent iff no vector in S is expressable as a linear combination of the other vectors in S.

Question 3

Theorem of linear dependence (3)

Answer 3

1) A finite set that contains 0 is linearly dependent.
2) A set with exactly one vector in slinearly independet if that vector is not 0
3) A set with with exactly two vectors is linearly independent iff neither vector is not a scalar multiple of the other.

Question 4

Theorem of whether the set S is linearly dependent

Answer 4

Let S = {v1, v2, …, vr} be a set of vectors in R^(n), if r>n then S is linearly dependent.

Question 5

Def: The Wronskian

Answer 5

If f1=f1(x), f2=f2(x), …, f2=f2(x) are functions that are at least n-1 times differentiable on thei nterval (-∞,∞), then the determinant of the matrix where r1 is the functions, r2 is the first derivative of the functions and rn is the (n-1)’th derivative of the functions is denoted W(x) and is called The Wronskian.

Question 6

Theorem of the Wronskian?

Answer 6

If the function of f1, f2, … , fn has (n-1) continous derivatives on the interval (-∞,∞), and if the wronskian of these functions is not identically zero on (-∞,∞), then these functions form a linearly independent set of vectors in C^(n-1)(-∞,∞)

++ The converse, of this theorem is false, if the functions are identically zero, then no conclusion can be reached about the linear independence.

Question 7

What does it mean that a function is identically zero?

Answer 7

A function identical to the zero function