To prove that $\vec{U} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ is a linear combination of the vectors $\vec{V}_1 = \begin{pmatrix} -0.5 \\ 2 \end{pmatrix}$ and $\vec{V}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, we need to find scalars $a$ and $b$ such that:

$\vec{U} = a \vec{V}_1 + b \vec{V}_2$

Substituting the given vectors into this equation, we get:

$\begin{pmatrix} 2 \\ -3 \end{pmatrix} = a \begin{pmatrix} -0.5 \\ 2 \end{pmatrix} + b \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

This equation can be written as a system of linear equations:

$\begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -0.5a + b \\ 2a + b \end{pmatrix}$

This gives us the following system of equations:

$-0.5a + b = 2 \quad \text{(1)}$

$2a + b = -3 \quad \text{(2)}$

We can solve this system using substitution or elimination. First, let's solve equation (1) for $b$:

$b = 2 + 0.5a$

Next, substitute this expression for $b$ into equation (2):

$2a + (2 + 0.5a) = -3$

Simplify and solve for $a$:

$2a + 2 + 0.5a = -3$

$2.5a + 2 = -3$

$2.5a = -5$

$a = -2$

Now, substitute $a = -2$ back into the expression for $b$:

$b = 2 + 0.5(-2)$

$b = 2 - 1$

$b = 1$

Therefore, the scalars $a = -2$ and $b = 1$ satisfy the system of equations. Thus, we can write:

$\vec{U} = -2 \vec{V}_1 + 1 \vec{V}_2$

So, $\vec{U} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ is indeed a linear combination of $\vec{V}_1 = \begin{pmatrix} -0.5 \\ 2 \end{pmatrix}$ and $\vec{V}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ with the coefficients $a = -2$ and $b = 1$.

Would you like more details on any part of this process or have any questions?

Here are some related questions:

1. What is a linear combination of vectors?
2. How do you solve a system of linear equations?
3. What is the significance of the coefficients in a linear combination?
4. How can you determine if a set of vectors is linearly independent?
5. Can a vector be a linear combination of non-basis vectors?
6. How does the dimension of a vector space relate to linear combinations?
7. What is the geometric interpretation of a linear combination of vectors?
8. What role do linear combinations play in vector spaces and subspaces?

Tip: When solving a system of linear equations, you can use methods like substitution, elimination, or matrix operations such as row reduction to find the solution efficiently.