Linear Algebra Section 1.5: Solutions of linear systems
Linear system is homogenous if
it is of the form Ax = 0 vector in matrix equation form.
Parametric vector equations (through a line and through a plane)
line: x = t(v) plane: x = su + tv
N(A)
solution set of Ax=0
When Ax=b has no solution,
the solution set is empty
Trivial solution
x = 0; of Ax = 0
homogenous linear system of size m by n has infinitely many solutions when...
...when n > m either infinitely many or exactly one solution when n = m
Why should A(cv + dw) = 0 for each pair of scalars "C" and "D"?
A(cv + dw) = = A(cv) + A(dw): Distribity property Rn = c(Av) + d(aw): Compatitbility property of Rn = c(0) + d(0) = 0 + 0 = 0
Is a homogenous solution consistent or inconsistent and why?
ALWAYS consistent because you can always write down the solution; talking all variable to be a zero vector.
Solutions of non-homogenous linear system
Ax = b, b ≠ 0
A is nonsingular if
N(A) = 0
A is singular if
N(A) = infinite
If b cannot equal zero, can Ax=b be a plane through the origin?
No. The equation of a plane through the origin has Ax=b and MUST = 0.
Equation of a plane through the origin:
Po = v1x + v2y + v3z = 0
THEOREM: Suppose Ax=b is consistent and p is a particular solution Hints: p, v, N(A)
The the solution set of Ax=b is the set of all vectors of the form: p + v where v ⋲ N(A) => p + N(A)
Does a homogenous linear system have a non-trivial solution? Why?
Yes because it has a free varibale
Let "A" be an m by n matrix and let "v" and "w" be vectors in Rn with Av = 0 and Aw = 0. Why must A(v+w) must be the zero vector?
a(v+w) = = av + aw = 0 + 0 = 0 This is the distributive property of Rn