← 2019 Paper 1

UPSC 2019 Maths Optional Paper 1 Q3c-ii — Step-by-Step Solution

5 marks · Section A

Rank and nullity; rank-nullity theorem · Linear Algebra · asked 9× in 14 yrs · Read the full method →

Question

Find the dimension of the subspace $V=\left\{(x_1,x_2,x_3,x_4)\in\mathbb R^4\ \Big|\ A\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}=0\right\}$.

(Here $A$ is the $4\times4$ matrix of part (c)(i).)

Technique

Rank–nullity theorem $\dim\ker A=n-\operatorname{rank}A$ with $n=4$, $\operatorname{rank}=2$.

Solution

Step 1 — $V$ is the null space; apply rank–nullity

$V=\ker A$. The rank–nullity theorem for $A:\mathbb R^4\to\mathbb R^4$ gives

From part (c)(i), $\operatorname{rank}(A)=2$, so

Answer