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?

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.