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UPSC 2019 Maths Optional Paper 1 Q1c — Step-by-Step Solution

10 marks · Section A

Matrix of a linear transformation · Linear Algebra · asked 10× in 13 yrs · Read the full method →

Question

Let $T:\mathbb R^2\to\mathbb R^2$ be a linear map such that $T(2,1)=(5,7)$ and $T(1,2)=(3,3)$. If $A$ is the matrix corresponding to $T$ with respect to the standard bases $e_1,e_2$, then find Rank $(A)$.

Technique

Recover the standard matrix via $A=[\text{images}]\,P^{-1}$; rank from a nonzero determinant. Faster: image of a basis is independent $\Rightarrow$ full rank.

Solution

Step 1 — Set up the equation for $A$

Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ be the standard matrix of $T$, so $T(v)=Av$ (columns). The data say

Stacking these as columns,

Step 2 — Solve for $A$

Let $P=\begin{bmatrix}2&1\\1&2\end{bmatrix}$, $\det P=4-1=3\neq0$, so $P$ is invertible and

Then

Step 3 — Rank of $A$

A $2\times2$ matrix with nonzero determinant is invertible, hence of full rank.

Answer