The geometry of solutions of systems of linear equations.
(a) Consider a linear equation of the form
a1x1 + · · · + anxn = b,
with a1, . . . , an, b ∈ R. Show that if (x1, . . . , xn)T and (y1, . . . , yn)T are two dis-
tinct solutions of this equation, then for every λ ∈ R,
(λx1 + (1 − λ)y1, . . . , λxn + (1 − λ)yn)T
is also a solution. What does it mean geometrically for the solution set?
(b) Use (a) to recover the fact that a system of linear equations has either zero, one,
or infinitely many solutions.
(c) Below are some subsets of R3. For each of them, explain why they cannot be
the set of solutions of a system of linear equations in three variables, or give an
example of a system of linear equations in three variables whose set of solutions
is the given subset: a line in R3; the union of two lines of R3; a plane in R3
.
(a) Consider a linear equation of the form
a1x1 + · · · + anxn = b,
with a1, . . . , an, b ∈ R. Show that if (x1, . . . , xn)T and (y1, . . . , yn)T are two dis-
tinct solutions of this equation, then for every λ ∈ R,
(λx1 + (1 − λ)y1, . . . , λxn + (1 − λ)yn)T
is also a solution. What does it mean geometrically for the solution set?
(b) Use (a) to recover the fact that a system of linear equations has either zero, one,
or infinitely many solutions.
(c) Below are some subsets of R3. For each of them, explain why they cannot be
the set of solutions of a system of linear equations in three variables, or give an
example of a system of linear equations in three variables whose set of solutions
is the given subset: a line in R3; the union of two lines of R3; a plane in R3
.
