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what are the correct answers/options to this question

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?

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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.

Let B = [[1, 2], [0, 3]], and let T: R^2 → R^2 be a linear transformation given by T(x) = Bx. Then T([1, 3]) is (choose all that applies):

- a linear combination of the columns of B
- a vector in R^2
- B[1, 3]
- [7, 9]

Explore how to determine the result of a linear transformation T(x) = Bx using matrix multiplication. This problem helps to understand key linear algebra concepts such as linear combinations, vectors in R^2, and matrix-vector products. Suitable for advanced high school and college students.

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