Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where:
$${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$
It's given that:
$${\left[ T \right]_{B \to E}} = \left( {\matrix{
1 & 1 \cr
1 & 3 \cr
} } \right)$$
I need to find what ${\left[ T \right]_E}$ is.
So, by definition of ${\left[ T \right]_{B \to E}}$:
$${\left[ T \right]_{B \to E}} = \left( {\matrix{ {T{{\left( {\matrix{ 1 \cr 0 \cr } } \right)}_B}} & {T{{\left( {\matrix{ 0 \cr 1 \cr } } \right)}_B}} \cr } } \right)$$
Hence, $$T{\left( {\matrix{ 1 \cr 0 \cr } } \right)_B} = \left( {\matrix{ 1 \cr 1 \cr } } \right),T{\left( {\matrix{ 0 \cr 1 \cr } } \right)_B} = \left( {\matrix{ 1 \cr 3 \cr } } \right)$$
Because $E$ is the standard basis, we deduce that:
$${\left[ T \right]_E} = \left( {\matrix{ 1 & 1 \cr 1 & 3 \cr } } \right)$$
Is that right? I am little bit confused by this material