## Basis with Respect to Which the Matrix for Linear Transformation is Diagonal

## Problem 315

Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by

\[T(ax+b)=(3a+b)x+a+3,\]
for any $ax+b\in P_1$.

**(a)** With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation $T$.

**(b)** Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

**(c)** Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.