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The image you uploaded contains five sections with various mathematical tasks in linear algebra, calculus, and analytical geometry. Below is a breakdown of the consigned problems:

### CONSIGNA 1:
1. **Part a:** Prove that the given set of vectors forms a basis and write vector \( \vec{W} \) in that basis.
   - Vectors: \( A = \{(1, -1), (2, 1)\} \)
   - Vector: \( \vec{W} = i + j \)

2. **Part b:** Find the volume of the tetrahedron determined by the vectors \( \vec{w} = i - 2j + k \), \( \vec{u} = i + j + k \), and \( \vec{v} = i + j - 2k \).

3. **Part c:** State and demonstrate the property when the scalar product of two vectors is zero.

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### CONSIGNA 2:
1. **Part a:** Decompose the given polynomial in \( \mathbb{C} \) and \( \mathbb{R} \) using the Gauss Theorem. Verify the hypotheses of the theorem and apply it.
   - Polynomial: \( p(x) = 13x^{10} + 75x^7 + 33x^4 + 66x^3 + 38x^9 + 57x^6 + 28x^5 + 8x^8 + 2x^{11} \)

2. **Part b:** State and prove the property of the complex roots of a polynomial with real coefficients.

---

### CONSIGNA 3:
1. **Part a:** Find the solution set for the system using Gauss-Jordan elimination and discuss the solution.
   \[
   \begin{aligned}
   2x + 2y - 2z &= 0 \\
   z &= 1 - 2x \\
   5z + 5y &= 1
   \end{aligned}
   \]

2. **Part b:** Solve the matrix equation \( \frac{1}{2}AXB = 2C \) for given matrices:
   \[
   A = \begin{pmatrix} 1 & 1 \\ -2 & 1 \end{pmatrix}, \, B = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}, \, C = \begin{pmatrix} 2 & 1 \\ -2 & -1 \end{pmatrix}
   \]

---

### CONSIGNA 4:
1. **Part a:** Find the equation of the line that passes through point \( P(2, 3) \) and forms an angle of \( \frac{\pi}{4} \) with the x-axis. Express it in all its forms.

2. **Part b:** Graph and express parametrically the following space line:
   \[
   \begin{aligned}
   2x + 2y - z &= 1 \\
   x + y + 2z &= 1
   \end{aligned}
   \]

---

### CONSIGNA 5:
1. **Part a:** Identify the type of geometric loci represented by the given equations and sketch them:
   - \( (x - 1)^2 - y^2 = 0 \)
   - \( (x + 1)^2 + 2y^2 = 0 \)

2. **Part b:** Analyze whether \( 2x^2 + y^2 + 4y = 0 \) is a pure or degenerate conic and graph it. Identify the type and obtain parametric equations.

3. **Part c:** For each of the following, identify if the equation is a plane, curve, space curve, or surface, determine the geometric name, and sketch it with justification:
   - \( x^2 + y^2 = 0 \)
   - \( z = y + x^2 \)
   - \( z^2 + y^2 - x = 0 \)
   - \( y^2 - x^2 = z^2 \)
   - \( x^2 + y^2 = z \)
   - \( z = \sin(y + x) \)
   - \( x + y + z = 1 \)
   - \( z^2 + y^2 = 1 \)
   - \( \frac{x}{4} + \frac{y}{4} + \frac{z}{1} = 1 \)

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Would you like me to solve one of these problems or explain any part in more detail? 

Here are 5 questions to guide the next steps:
1. Would you like assistance with Consigna 1a, proving the basis and writing the vector \( \vec{W} \)?
2. Should we solve the matrix equation in Consigna 3b using matrix multiplication?
3. Do you want help with Gauss-Jordan elimination from Consigna 3a?
4. Do you need help decomposing the polynomial in Consigna 2a?
5. Are you looking for assistance in identifying and sketching the loci in Consigna 5a?

**Tip:** When working with matrices, always check if they are invertible before attempting certain operations like multiplication or solving equations.

The user uploaded a set of mathematical problems covering topics in linear algebra, calculus, and analytical geometry.

This page provides solutions for advanced mathematical problems including linear algebra basis, Gauss-Jordan elimination, matrix equations, and conic section analysis. Topics such as polynomial decomposition and properties of complex roots are also covered. Suitable for university-level students.

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