prove that $\vec{U} = \binom{2}{-3}$ is a linear combination of the vectors $\vec{V}_1 = \binom{-0.5}{2}$ and $\vec{V}_2 = \binom{1}{1}$,

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.

Learn how to prove that vector U = (2, -3) is a linear combination of vectors V1 = (-0.5, 2) and V2 = (1, 1) using methods like substitution and elimination. This problem covers key concepts in linear algebra, including vector spaces and solving systems of linear equations. Suitable for undergraduate students studying linear algebra.

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