Chapter D, Problem 2P

a.

Program Plan Intro

To prove that $\left|R\left(A\right)\right|=2^r$ and conclude that A defines a permutation on $S_n$ only if A has full rank.

Given information:

• r is the rank of matrix A.
• Preimage of y is defined by $P\left(A,y\right)=\left\{x:Ax=y\right\}$ , for given $n\times n$ matrix A and given a value $y\in R\left(A\right)$ .

Explanation:

It can be assumed without generality loss, that the first r columns of A are linearly independent.

Then for $x_1,x_3\in S_n$ , it can be concluded that $A{x}_1\ne A{x}_2$ based on following two conditions.

• In the first r entries $x_1$ and $x_2$ are not identical. And $x_1$ , $x_2$
• have 0’s in the remaining entries.

Now $A{x}_1\ne A{x}_2$ is true since the first r entries of each are a linear combination of the first r rows of A , and it is impossible to have two different linear combinations that are equal of them because they are independent.

Now it is must that $\left|R\left(A\right)\right|\ge 2^r$ as there are at least $2^r$ non-equivalent vectors $x\in S_n$ . Alternatively, x is a vector that doesn’t consists of 0’s in the coordinates that larger than r . Therefore $Ax={\displaystyle \sum _{x_i}{a}_i}$ where a_i is the $i^{th}$ column of A . Now as each of the last $n-r$ columns of A is in fact a linear combination of the first r columns of A , this can be rewritten as a linear combination of the first r columns of A .

Since all of these has been already, $\left|R\left(A\right)\right|=2^r$ . The range cannot include all $2^n$ elements of $S_n$ if A doesn’t have full rank so, permutation can be defined possible by A

b.

Program Plan Intro

To prove that $\left|P\left(A,y\right)\right|=2^{n-r}$ if r is the rank of $n\times n$ matrix A and $y\in R\left(A\right)$ .

c.

Program Plan Intro

To prove that $\left|B\left(S^{\text{'}},m\right)\right|=2^r$ and that for each block in $B\left(S^{\text{'}},m\right)$ exactly $2^{m-r}$ numbers in S map to that block.

d.

Program Plan Intro

To show that number of linear permutations of $S_n$ is much less than number of permutations of $S$ using counting argument.

e.

Program Plan Intro

To provide an example of a value n and permutation of $S_n$ that cannot be achieved by any linear permutation.