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 $B’$.

Let $A=(a_{ij})$ be an $n \times n$ matrix.
We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have
\[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\]
for $1 \leq i, j \leq n$.

Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.

(a)The stochastic matrix $A$ has an eigenvalue $1$.

(b) The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.