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.

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.

1

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