if x = (-2, -1) in cartesian coordinates, how is it expressed with the basis vectors v1=(1, -1) and v2=(1, 1)

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.

If x = (-2, -1) in Cartesian coordinates, how is it expressed with the basis vectors v1 = (1, -1) and v2 = (1, 1)?

Learn how to express the Cartesian point (-2, -1) as a linear combination of the basis vectors v1 = (1, -1) and v2 = (1, 1). This problem involves solving a system of equations and understanding basis transformations in linear algebra. Ideal for high school advanced math or undergraduate students.

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