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UPSC 2016 Maths Optional Paper 1 Q2a-ii — Step-by-Step Solution

6 marks · Section A

Matrix of a linear transformation · Linear Algebra · asked 10× in 14 yrs · Read the full method →

Question

If $T:P_2(x)\to P_3(x)$ is such that $T(f(x))=f(x)+5\displaystyle\int_0^x f(t)\,dt$, then choosing $\{1,1+x,1-x^2\}$ and $\{1,x,x^2,x^3\}$ as bases of $P_2(x)$ and $P_3(x)$ respectively, find the matrix of $T$.

Technique

Apply the operator to each domain basis polynomial, then write its coordinates in the codomain basis; here the codomain basis is standard, so coordinates are coefficients.

Solution

Step 1 — Apply $T$ to each domain basis polynomial

$T(f)=f+5\int_0^x f\,dt$.

$f=1$: $\int_0^x 1\,dt=x$, so $T(1)=1+5x$.

$f=1+x$: $\int_0^x(1+t)\,dt=x+\tfrac{x^2}{2}$, so $T(1+x)=(1+x)+5\big(x+\tfrac{x^2}{2}\big)=1+6x+\tfrac52 x^2$.

$f=1-x^2$: $\int_0^x(1-t^2)\,dt=x-\tfrac{x^3}{3}$, so $T(1-x^2)=(1-x^2)+5\big(x-\tfrac{x^3}{3}\big)=1+5x-x^2-\tfrac53 x^3$.

Step 2 — Coordinates in the codomain basis $\{1,x,x^2,x^3\}$

Since the codomain basis is the standard one, coordinates are just the coefficients:

Step 3 — Assemble the matrix (columns = images)

Answer