https://file.aiquickdraw.com/mathbot/file/1726911788811bclcfufc.jpg

I see that you’ve uploaded an image from the book *Linear Algebra Done Right* by Sheldon Axler, specifically from the section on span and linear independence (Exercises 2A). Here are some of the questions presented in the image:

1. Find a list of four distinct vectors in \( \mathbb{F}^3 \) whose span equals \( \{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0 \} \).

2. Prove or give a counterexample: If \( v_1, v_2, v_3, v_4 \) spans \( V \), then the list \( v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4 \) also spans \( V \).

3. Suppose \( v_1, \dots, v_m \) is a list of vectors in \( V \). For \( k \in \{1, \dots, m \} \), let \( w_k = v_1 + \dots + v_k \). Show that \( \text{span}(v_1, \dots, v_m) = \text{span}(w_1, \dots, w_m) \).

4. 
   (a) Show that a list of length one in a vector space is linearly independent if and only if the vector in the list is not 0.
   (b) Show that a list of length two in a vector space is linearly independent if and only if neither of the two vectors in the list is a scalar multiple of the other.

5. Find a number \( t \) such that \( (3, 1, 4), (2, -3, 5), (5, 9, t) \) is not linearly independent in \( \mathbb{R}^3 \).

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Would you like to focus on a particular problem from this list or go over a solution in detail? 

Here are some related questions to expand the concepts:

1. What are the conditions for a set of vectors to span a vector space?
2. How do you check if a set of vectors is linearly independent?
3. What is the geometric interpretation of spanning in \( \mathbb{R}^3 \)?
4. How does changing a single vector in a set affect the linear independence of the set?
5. Can you provide an example of a set of linearly dependent vectors in \( \mathbb{R}^3 \)?

**Tip**: To prove linear independence, remember that a set of vectors is linearly independent if and only if the only solution to the equation \( c_1 v_1 + c_2 v_2 + \dots + c_n v_n = 0 \) is \( c_1 = c_2 = \dots = c_n = 0 \).

Find a list of four distinct vectors in F^3 whose span equals {(x, y, z) ∈ F^3 : x + y + z = 0}.

This linear algebra problem asks to find four distinct vectors in F^3 whose span equals {(x, y, z) ∈ F^3 : x + y + z = 0}. It requires knowledge of span, vector spaces, and linear independence.

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