Question

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V \ W and v ∈ V \ W, then u + v ∈ V \ W.
Which one of the following holds?

1. Both P and Q are true
2. P is true but Q is false
3. P is false but Q is true
4. Both P and Q are false

Answer: Both P and Q are false

Solution :-

Statement P :

Statement ‘P’ is not always true. Lets consider an example of system of linear equations

Now, lets solve this system of linear equations. Here,

Here, Matrix A is 3 × 2 and b is 3 × 1.

Hence, augmented matrix is

Applying $R_2\to R_2+\left(-2\right)R_1$ and $R_3\to R_3+3R_1$ , then we get

Applying $R_2\to -R_2$ , then we get

Hence, $x_1-2x_2=1\cdots \cdots \left(1\right)$ and $x_2=-1$
put the value of $x_2=-1$ in eq(1), we get $x_1=-1$

So, $x_1=-1$ and $x_2=-1$ is a solution of the system of linear equations and we can see that this solution is unique but $3\ne 2$ . Here, 3 is the number of rows of A and 2 is the number of columns of A.

Therefore, If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n is not always true, So Statement ‘P’ is false.

Statement Q :

Statement ‘Q’ is also not always true. lets us consider vector space V and subspace W.

$W=\{\left(x,y,0\right):x,y\in ℝ\}$ , Here we can see that W is subspace of V.

Now, $V\backslash W=\{\left(x,y,z\right):x,y,z\in ℝ;z\ne 0\}$ . Now, we take two vector from $V\backslash W$ , $u=\left(x,y,z\right)\in V\backslash W,v=\left(x,y,-z\right)\in V\backslash W\mathrm{.}$
Now, $u+v=\left(x,y,z\right)+\left(x,y,-z\right)=\left(2x,2y,0\right)\in W$ , not in $V\backslash W$ .

Hence, Statement ‘Q’ For a subspace W of a nonzero vector space V, whenever u ∈ V \ W and v ∈ V \ W, then u + v ∈ V \ W is false. Here, ‘whenever’ word is important.

So, the correct answer is option-4, $\to$ Both P and Q are false