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The document you uploaded appears to be a linear algebra assignment from the Instituto Federal de Mato Grosso do Sul (IFMS). Here are the main problems listed in the assignment:

1. **Subspaces of \( \mathbb{R}^4 \)**: 
   - Show that the given sets in \( \mathbb{R}^4 \) are subspaces.
     - \( w = \{(x, y, z, t) \in \mathbb{R}^4 \mid x + y = 0 \text{ and } z - t = 0 \} \)
     - \( w = \{(x, y, z, t) \in \mathbb{R}^4 \mid 2x + y - t = 0 \text{ and } z = 0 \} \)

2. **Subspace of \( M_2 \)**:
   - Consider the subspace \( w \) of \( M_2 \) defined by:
     \[
     w = \left\{ 
     \begin{pmatrix}
     2a & a + 2b \\
     0 & a - b 
     \end{pmatrix} \mid a, b \in \mathbb{R}
     \right\}
     \]
   - Determine whether certain matrices belong to \( w \) and find a basis for \( w \).

3. **Linear Applications**:
   - Determine which of the given functions are linear transformations.

4. **Linear Transformation**:
   - Find the linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) with given properties and determine a specific vector.

5. **Subspace of \( \mathbb{R}^4 \) generated by vectors**:
   - Consider the subspace generated by given vectors and answer questions related to linear independence, bases, and dimension.

6. **Linear Transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^3 \)**:
   - Determine a linear transformation with specified properties and find the image of given vectors.

7. **Transformation \( T(x, y, z) = (z, x - y, -z) \)**:
   - Analyze the kernel, image, surjectivity, and inverse of the transformation.

If you have any specific questions or need detailed explanations for any of these problems, feel free to ask!

Here are 5 related questions you might explore:

1. How do you determine if a set is a subspace of a vector space?
2. What is the significance of finding a basis for a subspace?
3. How can you verify if a function is a linear transformation?
4. What are the steps to find the kernel and image of a linear transformation?
5. How do you determine if a linear transformation is surjective?

**Tip:** When dealing with subspaces, always check if the set satisfies the three criteria: containing the zero vector, closure under addition, and closure under scalar multiplication.

The document you uploaded appears to be a linear algebra assignment from the Instituto Federal de Mato Grosso do Sul (IFMS).

Explore the key concepts of linear algebra through this assignment, covering subspaces in R4, basis and dimension, and linear transformations. This content is designed for first-year undergraduate students.

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