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The assignment contains problems on Linear Algebra and Calculus. Here's a summary of the tasks:

### Part I: Linear Algebra
1. **Solve a system of equations** using the Gauss elimination method.
2. **Find the system of linear equations** whose solution set represents a subspace spanned by the vectors \( a_1 = (0,1,1,-1), a_2 = (1,0,1,1), a_3 = (1,1,0,2) \).
3. **Subspace problems**: Find \( L_1 + L_2 \) and \( L_1 \cap L_2 \), where:
   - \( L_1 \) is spanned by \( a_1 = (1,2,1), a_2 = (1,-1,1), a_3 = (3,3,1) \)
   - \( L_2 \) is spanned by \( b_1 = (-1,3,2), b_2 = (2,2,1), b_3 = (-3,1,1) \)

### Part II: Calculus
1. **Simplify a partial differential equation** by introducing new variables \( \xi = x + y, \eta = x - y \).
2. **Prove the existence of a unique function** \( y(x) \) satisfying an equation \( x = y + \phi(y) \) around \( x = 0 \), with initial condition \( y(0) = 0 \).
3. **Find the directional derivative** of a function \( f(x, y, z) = x / \sqrt{x^2 + y^2 + z^2} \) at point \( (1,2,-2) \) in the direction of a unit vector tangent to a parametric curve.

Would you like detailed solutions for any of these problems? Here are some related questions that might help your understanding:

1. What are the steps for solving systems using the Gauss elimination method?
2. How can you find the system of equations representing a linear subspace?
3. What are the general methods for finding the sum and intersection of subspaces?
4. How does introducing new variables simplify partial differential equations?
5. What techniques are used to prove the existence and uniqueness of a solution to differential equations?

**Tip**: In Gauss elimination, it's important to perform row operations carefully to maintain the consistency of the system while converting it to row-echelon form.

Solve problems on Linear Algebra and Calculus. The problems include solving a system of equations using Gauss elimination, finding a linear subspace spanned by vectors, performing operations with subspaces, simplifying a partial differential equation, proving the existence and uniqueness of a solution to a differential equation, and finding the directional derivative of a function in 3D space.

This page provides step-by-step solutions to linear algebra and calculus problems, including solving systems of equations via Gauss elimination, finding linear subspaces, simplifying partial differential equations, and proving the existence and uniqueness of solutions. Suitable for undergraduate economics and mathematics students.

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