What is the number of linearly independent solutions of a set of vectors taken from identity matrix?

Question

If I have 3 sets of vectors X=(1,0,0) , Y= (0,1,0) and Z=(0,0,1) then these all form a linearly independent set of vectors ,so K(X)+P(Y)+Q(Z) =0 , so we get K=P=Q=0 , but my confusion is what is the no of independent solutions in this , since rank of matrix formed from these vectors will be 3 and according to the formula No of independent solutions =no of unknowns- rank of matrix , we get no of independent solutions =0 , so how can it be 0 here ?

Answer 1

A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the other vectors. 

K₁X₁+K₂X₂+K₃X₃=0

X₁ = - K₂/K₁ X₂ - K₃/K₁ X₃

_and If no vector in the set can be written in this way, then the vectors are said to be linearly independent. 

K₁X₁ + K₂X₂ + K₃X₃=0

implies K₁=K₂=K₃=0

Given

X(1,0,0)   Y(0,1,0)  Z(0,0,1)

K₁(1,0,0) +K₂(0,1,0) + K₃(0,0,1)=(0,0,0)

(K₁,0,0) +(0,K₂,0) + (0,0,K₃)=(0,0,0)

i.e. K₁=K₂=K₃=0

So vectors are linearly independent. 

Here Rank =3

So and number of linearly independent.  solution = no of variables -  Rank of the matrix

                                                                          = 3-3 =0

So there There is no linearly independent. 

zero Solution will be the only solution which is Trivial.

Comments

How  number of linearly independent.  solution = no of variables -  Rank of the matrix?

Can you give me some intuition over this formula?