← 2022 Paper 1

UPSC 2022 Maths Optional Paper 1 Q1b — Step-by-Step Solution

10 marks · Section A

Linear transformations · Linear Algebra · asked 2× in 13 yrs · Read the full method →

Question

Let $T:\mathbb R^2\to\mathbb R^3$ be a linear transformation such that $T(1,0)^T=(1,2,3)^T$ and $T(1,1)^T=(-3,2,8)^T$. Find $T(2,4)^T$.

Technique

Standard linearity reconstruction; recover $T(e_1),T(e_2)$ from the two given values, then apply matrix form.

Solution

Strategy. Express $(2,4)^T$ as a linear combination of $(1,0)^T$ and $(1,1)^T$. Use linearity.

Step 1 — Find $T(0,1)^T$

$(1,1)=(1,0)+(0,1)$, so $(0,1)^T=(1,1)^T-(1,0)^T$. By linearity:

Step 2 — Decompose $(2,4)^T$

$(2,4)^T=2(1,0)^T+4(0,1)^T$.

Step 3 — Apply linearity

Compute:

• $2(1,2,3)=(2,4,6)$.
• $4(-4,0,5)=(-16,0,20)$.
• Sum: $(2-16,\;4+0,\;6+20)=(-14,4,26)$.

Answer