CS972: Assignment 1
April, 2023
Time: 3 days Maximum Marks: 50
Question (Full marks = 50 ) Consider the set of solutions S of a linear equation:
S = {(x1, x2, . . . , xn) ∈ Qn | a1x1 + a2x2 + · · · + anxn = 0}.
Here the Q is the set of rational numbers and a1, a2, . . . , an ∈ Q.
1. Defne a linear function f : Qn 7→ Q such that S is precisely the null space of f. Show
that the dimension of S is n – 1 if not all ai’s are 0. [5+10]
Now consider a collection of m linear equations:
a1,1x1 + a1,2x2 + · · · + a1,nxn = 0
a2,1x1 + a2,2x2 + · · · + a2,nxn = 0
…
…
a
m,1x1 + am,2x2 + · · · + am,nxn = 0
with ai,j ∈ Q. Let S ⊆ Qn be the set of solutions of these equations.
2. Defne a linear function f : Qn 7→ Qm such that S is precisely the null space of f.
Obtain a matrix representation F of f. [5+5]
3. Let F′ be the matrix obtained by doing Gaussian elimination on the columns of F.
Show that the null space of F′ is also S. [10]
4. Computation of F′ allows us to easily fnd solutions of the collection given. Show how
to use F′ to fnd a basis for vector space S. [15]
1