 For which values x, vectors a and b are basis of some space? Explain
 your answer.
 a = 1−x
 x ,b= 1−x

To determine when vectors \( \mathbf{a} \) and \( \mathbf{b} \) form a basis of some space, we need to examine the linear independence of these vectors.

### Step 1: Define Vectors \( \mathbf{a} \) and \( \mathbf{b} \)
Given:
\[
\mathbf{a} = \begin{bmatrix} 1 - x \\ x \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 1 - x \\ 1 \end{bmatrix}.
\]

### Step 2: Conditions for Linear Independence
For two vectors in \(\mathbb{R}^2\) to be a basis, they must be **linearly independent**. Vectors are linearly independent if no scalar \(c\) exists (except \(c = 0\)) such that \( \mathbf{a} = c \mathbf{b} \).

Alternatively, we can use the **determinant** of the matrix formed by placing these vectors as columns:
\[
\text{det} \left( \begin{bmatrix} 1 - x & 1 - x \\ x & 1 \end{bmatrix} \right).
\]
If the determinant is non-zero, the vectors are linearly independent.

### Step 3: Compute the Determinant
Let's calculate the determinant of the matrix:
\[
\text{det} \left( \begin{bmatrix} 1 - x & 1 - x \\ x & 1 \end{bmatrix} \right) = (1 - x) \times 1 - (1 - x) \times x.
\]
Simplifying this, we get:
\[
\text{det} = 1 - x - x + x^2 = x^2 - 2x + 1.
\]

### Step 4: Find the Values of \( x \) Where Determinant is Zero
Set the determinant to zero to find the values of \( x \) for which the vectors are linearly dependent:
\[
x^2 - 2x + 1 = 0.
\]
This is a quadratic equation. Factoring, we get:
\[
(x - 1)^2 = 0.
\]
Thus, the only solution is:
\[
x = 1.
\]

### Conclusion
The vectors \( \mathbf{a} \) and \( \mathbf{b} \) will be linearly independent (and hence form a basis) for **all values of \( x \) except \( x = 1 \)**. When \( x = 1 \), the vectors become linearly dependent, meaning they do not form a basis of the space.

Would you like more details or have any questions?

### 5 Related Questions to Explore Further
1. What does it mean for a set of vectors to form a basis in \( \mathbb{R}^n \)?
2. How can the concept of linear independence be extended to more than two vectors?
3. What is the geometric interpretation of linear dependence in two-dimensional space?
4. How do you determine if three vectors in \( \mathbb{R}^3 \) form a basis?
5. What role does the determinant play in understanding vector independence?

### Tip
When checking if vectors form a basis in \(\mathbb{R}^n\), remember they must both span the space and be linearly independent. The determinant is a quick way to test linear independence for small sets of vectors.

For which values x, vectors a and b are basis of some space? Explain your answer.
 a = 1−x x ,b= 1−x

Discover how to determine for which values of x the vectors a = [1 - x, x] and b = [1 - x, 1] form a basis of a vector space. Explore linear independence using the determinant and solve for x. Suitable for advanced high school or college students studying linear algebra.

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