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Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of $P$ is __________
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These question can be solved using rank-nullity theorem, rank+nullity=no of variables.here they have mentioned all solution of the equation Px=0 are the scalar multiple of [2 5 4 3 1]^T.so,we can understand that n-1 solutions are dependent while one and only one solution is independent.And by the definition, nullity=number of independent solution,so nullity=1.no of variables is nothing but the no of columns in owr 4×5 matrix which is 5. Therefore rank=5-1=4.hope these is helpful.
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1st thing: Rank + Nulity = No. of unknowns (no. of columns )

Nulity refers Nullspace of the real m X n matrix

Nullspace contains those vectors which satisfies the Homogeneous equation

 

i.e If A is the Matrix then Nullspace of A will have vector(s) “x” such that Ax=0

Evetually Nullspace of  a Matrix = Span(all vectors “x”)

Coming to the Question,

Here the x (vector(s)) are all Lr.dependent , 

So, Null space = 1

=nulity =1

Rank = 5-1

check this: https://brilliant.org/wiki/rank-nullity-theorem/

 

 

 

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6 Answers

9 votes
9 votes
Best answer
Every solution to $Px = 0$ is scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$, It means out of 5 column vectors of matrix $P$,  $4$  are linearly independent as we have only one line in NULL Space (along the given vector).

Rank is nothing but the number of linearly independent column vectors in a matrix which is $4$ here.
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how can you say that one of the column vectors linearly dependent on remaining vectors
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edited by

 

If  we consider $P=\left \{ {C_1},{C_2},{C_3},{C_4},{C_5}\right \}$ and $X=\begin{bmatrix} x_1 &x_2 & x_3 & x_4 & x_5 \end{bmatrix}^T$

Since $P X = 0$

Then, $x_1C_1+x_2C_2+x_3C_3+x_4C_4+x_5C_5=0$

$\left ( Given, x_1=2k, x_2=5k, x_3=4k,x_4=3k,x_5=k \right )$

So, $2C_1+5C_2+4C_3+3C_4+C_5=0$

$C5=-2C_1-5C_2-4C_3-3C_4$

This proves that one column vector is dependent on the other $4$ independent column vectors.

Rank = number of linearly independent column vectors in a matrix which is $4$ here.


 

Other methods to solve this problem:

The dimension of the null space of matrix P is called the nullity of P.

Let say,  $u=\begin{bmatrix} 2 &5 &4 & 3 & 1 \end{bmatrix}^T$

Since every other solution can be expressed in the form of given vector, so we can write $X=k\cdot u$. This means u form the basis of null$(P)$ and null space has dimension 1. Hence nullity$(P)=1$,

Or,

nullity is the no of free variables in the solution of $PX=0$.

Here, $X=\begin{bmatrix} 2k \\ 5k \\ 4k \\ 3k \\k \end{bmatrix}$

Clearly, $x_1,x_2,x_3,x_4$ are dependent on $x_5$. Only 1 free variable in solution.

so nullity$(P)$ =1.

 


By rank-nullity theorem,

rank$(P)+$ nullity$(P)=$ no of column of P

$r+1=5 \rightarrow  r=4$

(Correct me if something wrong).

 

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What happens if the solution of the equation Px=0 is,

  1. a scalar multiple of [ 2 5 4 2 1]^T?
  1. a scalar multiple of [ 2 5 4 2 2]^T?
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Since every other solution can be expressed in the form of a given vector u, so we can write $X=k⋅u. $ (where k is scalar)

This means u form the basis of null(P) and null space has dimension 1. Hence $nullity(P)=1$,

$Rank = 5(no of column vectors)- 1(nullity)=4 $
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8 votes
8 votes
it is in form PX=0.

P is of order 4X5.  4 EQUATIONS AND 5 UNKNOWNS(5 variables)

n=5.

solution of PX=0 is

k  [2 5 4 3 1] transpose

here we have only 1 arbitary value.(that is only 1 scalar i.e.,K)

no. of arbitary values(scalars)= n-r

1=5-r

r=4
7 votes
7 votes
Rank $+$ Nullity = Number of Columns

Here, Nullity is $1$. (Nullity is the dimension of the null space)

Rank : $5\ – 1 = 4$

4 Comments

Very well explained. But I have one suggestion - along with “Nullity is the dimension of the null space”, you can also mention the fact that “Nullity of a matrix A is no. of possible column vectors X satisfying the equation AX = 0”. It will be more intuitive for some people.

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how nullity is one
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@Anuj,

Nullity is the dimension of the null space

in other words,

Nullity of a matrix A is no. of possible column vectors X satisfying the equation AX = 0

In the given question, we are interested in finding the nullity of P. So we need to find out the number of possible column vectors X satisfying the equation PX = 0. In the question, it is also given that

every solution of the equation PX=0 is a scalar multiple of [25431]$^T$

this means that X is of the form k [25431]$^T$ where k is a scalar. Clearly, the Null Space of P contains only 1 vector. Hence Nullity is 1.

You can also refer to this video for better understanding.

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6 votes
6 votes

This problem is based on Linear Homogeneous equations (System of Linear Equations)

1 comment

simple and nice explanation
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