Given the Cartesian coordinates $x = (-2, -1)$, we need to express this point as a linear combination of the given basis vectors $v_1 = (1, -1)$ and $v_2 = (1, 1)$.

Let the expression for $x$ in terms of $v_1$ and $v_2$ be:

$x = c_1 v_1 + c_2 v_2$

This means we want to solve for the scalars $c_1$ and $c_2$, where:

$(-2, -1) = c_1 (1, -1) + c_2 (1, 1)$

Writing this as a system of equations:

$-2 = c_1 (1) + c_2 (1) \quad \text{(for the x-component)}$ $-1 = c_1 (-1) + c_2 (1) \quad \text{(for the y-component)}$

System of equations:

1. $c_1 + c_2 = -2$
2. $-c_1 + c_2 = -1$

We can solve this system by adding both equations:

$(c_1 + c_2) + (-c_1 + c_2) = -2 + (-1)$ $2c_2 = -3$ $c_2 = -\frac{3}{2}$

Substitute $c_2 = -\frac{3}{2}$ into the first equation:

$c_1 + \left(-\frac{3}{2}\right) = -2$ $c_1 = -2 + \frac{3}{2} = -\frac{4}{2} + \frac{3}{2} = -\frac{1}{2}$

Thus, the coordinates of $x$ in terms of the basis vectors $v_1$ and $v_2$ are:

$x = -\frac{1}{2} v_1 - \frac{3}{2} v_2$

Or more clearly:

$x = \left( -\frac{1}{2}, -\frac{3}{2} \right) \quad \text{in the basis} \ \{v_1, v_2\}$

Would you like more details or clarifications?

Related questions:

1. How can we verify that these coordinates are correct?
2. What is the geometric interpretation of changing basis?
3. How do we find the inverse of a 2x2 matrix in the case of basis transformation?
4. What if the vectors $v_1$ and $v_2$ were not linearly independent?
5. How would this process change in higher dimensions, say for 3D space?

Tip:

When expressing a vector in a new basis, you're essentially reinterpreting the vector in terms of a different set of directions, which can simplify certain calculations depending on the problem.