CS972: Assignment 1
October 30, 2022
Time: 3 days Maximum Marks: 60
Question (10 × 6 = 60 marks) Consider the set of solutions S of a linear equation:
S = {(x1, x2, . . . , xn) ? Q
n
| a1x1 + a2x2 + · · · + anxn = 0}.
Here the Q is the set of rational numbers and a1, a2, . . . , an ? Q.
Define 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.
Now consider a collection of m linear equations:
a1,1x1 + a1,2x2 + · · · + a1,nxn = 0
a2,1x1 + a2,2x2 + · · · + a2,nxn = 0
.
.
.
.
.
.
am,1x1 + am,2x2 + · · · + am,nxn = 0
with ai,j ? Q. Let S ? Qn be the set of solutions of these equations. Define a linear
function f : Qn
7? Qm such that S is precisely the null space of f. Obtain a matrix
representation F of f.
Let F
0 be the matrix obtained by doing Gaussian elimination on the columns of F. Show
that null space of F
0
is also S.
Computation of F
0 allows us to easily find solutions of the collection given. Show how to
use F
0
to find a basis for vector space S.
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