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System of Linear Equations Theoretical

5 important questions on System of Linear Equations Theoretical

What is a system of m linear equations with n unknowns over field F?

It is a system of linear equations of the form
a11x1+a12x2+...+a1nxn=b1
a21x1+a22x2+...+a2nxn=b2
am1x1+am2x2+...+amnxn=bm
where a11,...,amn,b1...,bm are scalars from the field F and x1,...,xn are n variables taking values in F.

What is the coefficient matrix of the system?

-It is an mxn matrix made of a11,a12,a13,....amn
-If we denote x to be a column vector made of x1,x2,...xn
and we denote b to be another column vector made of b1,b2...,bm
-Then we can write the system as Ax=b
-When we get more than one results. We can denote s (column vector) for set of solutions of the system.
-If s is nonpempty, the system is consistent, otherwise the system is inconsistent.

What is a homogeneous system?

-A system of linear equations of the form Ax=b is homogeneous when b=0, otherwise it is nonhomogeneous.
-If we have Ax=0,m equations n unknowns over ,
and we denote K to be the set of all solutions to Ax=0 ,
then K=N(La) as K is a subspace of F
with dimension n-rank(La)=n-rank(A)
-If we take K to be the solution set of consistent system of equations Ax=b,
and Kh to be the solution set of homogeneous system Ax=0,
then K={s}+Kh={s+k: k in Kh}

What is a system of n linear equations with n unknowns over field F?

-If we have Ax=b and A is inver.tible then the system has only one solution A-1(b)
-If the system has only one solution then A is invertible

When is a system consistent?

-If we have Ax=b, then system is consistent iff rank(A)=rank(A|b)
(A|b) is the augmented matrix of A
-It is equivalent to say that dim(span({a1,a2,...,an}))=dim(span({a1,a2,...an,b}))

The question on the page originate from the summary of the following study material:

Elementary Linear Algebra (Classic Version) | 9780134689470 | Lawrence E Spence, et al

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