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To show that the vector \( w = (-2, 1) \) is a linear combination of vectors \( u = (1, 2) \) and \( v = (0, -3) \), we need to find scalars \( a \) and \( b \) such that:

\[
w = a \cdot u + b \cdot v
\]

This means:

\[
(-2, 1) = a \cdot (1, 2) + b \cdot (0, -3)
\]

Now, let's break this into two components (for the x- and y-coordinates):

### Step 1: Equation for the x-coordinate
\[
-2 = a \cdot 1 + b \cdot 0
\]
\[
-2 = a
\]

Thus, \( a = -2 \).

### Step 2: Equation for the y-coordinate
\[
1 = a \cdot 2 + b \cdot (-3)
\]
Substitute \( a = -2 \):
\[
1 = (-2) \cdot 2 + b \cdot (-3)
\]
\[
1 = -4 - 3b
\]
\[
3b = -4 - 1 = -5
\]
\[
b = \frac{-5}{-3} = \frac{5}{3}
\]

### Conclusion
The vector \( w = (-2, 1) \) can be written as a linear combination of \( u = (1, 2) \) and \( v = (0, -3) \) as follows:

\[
w = -2 \cdot u + \frac{5}{3} \cdot v
\]

This shows that \( w \) is a linear combination of \( u \) and \( v \).

Would you like further clarification on any part of the process?

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Here are 5 related questions to consider:

1. What are the conditions for a set of vectors to form a basis for \( \mathbb{R}^2 \)?
2. How can you determine if three vectors in \( \mathbb{R}^3 \) are linearly independent?
3. What is the geometrical interpretation of a linear combination in 2D space?
4. How do you solve for linear combinations in higher dimensions, such as \( \mathbb{R}^3 \) or \( \mathbb{R}^n \)?
5. What role do linear combinations play in determining the span of a set of vectors?

**Tip:** In vector spaces, finding linear combinations helps determine if one vector can be expressed using others, which is key to understanding the concept of vector span and independence.

Show that the vector w = (-2, 1) is a linear combination of vectors u = (1, 2) and v = (0, -3) in the vector space R^2.

Learn how to express the vector w = (-2, 1) as a linear combination of u = (1, 2) and v = (0, -3) in R^2. This detailed solution involves key concepts in linear algebra, such as vector spaces and the linear combination theorem. Suitable for high school students (Grades 10-12).

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