Given that $B = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ and the vector $\vec{x} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$, the linear transformation $T(\vec{x}) = B \vec{x}$ results in:

$T\left(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\right) = \begin{bmatrix} 7 \\ 9 \end{bmatrix}$

Now, let’s review the options:

• a linear combination of the columns of $B$: Correct, because $T(\vec{x})$ is essentially the linear combination of the columns of $B$ with the entries of $\vec{x}$ as weights.
• a vector in $\mathbb{R}^2$: Correct, since $T(\vec{x}) = \begin{bmatrix} 7 \\ 9 \end{bmatrix}$ is a vector in $\mathbb{R}^2$.
• $B \begin{bmatrix} 1 \\ 3 \end{bmatrix}$: Correct, this represents the same transformation as $T(\vec{x})$, which equals $\begin{bmatrix} 7 \\ 9 \end{bmatrix}$.
• $\begin{bmatrix} 7 \\ 9 \end{bmatrix}$: Correct, as this is the resulting vector from the transformation.

Thus, the correct answers are all of the options.

Would you like further explanation on any specific part?

Here are 5 related questions:

1. What is the definition of a linear transformation?
2. How do you compute a matrix-vector product?
3. What does it mean for a vector to be a linear combination of columns of a matrix?
4. How can we confirm that $\begin{bmatrix} 7 \\ 9 \end{bmatrix}$ is indeed in $\mathbb{R}^2$?
5. How do the properties of matrix multiplication apply to linear transformations?

Tip: When dealing with linear transformations, always visualize them as transformations in space, altering the magnitude and direction of vectors in a consistent way.