The image presents a vector space $V = \mathbb{R}^2$ and a set $S$ consisting of three vectors:

$S = \{ (1,2), (2,3), (3,4) \}.$

Understanding the Problem:

The given vectors belong to $\mathbb{R}^2$, which means they are 2-dimensional vectors. This kind of setup typically raises questions such as:

1. Is the set $S$ linearly independent?
2. Does the set $S$ form a basis for $\mathbb{R}^2$?
3. What is the span of $S$?

Key Observations:

1. There are 3 vectors in $S$, but the space $\mathbb{R}^2$ has dimension $2$. Therefore:
   • If the vectors in $S$ are linearly independent, they span at most a 2-dimensional subspace, and $S$ cannot form a basis for $\mathbb{R}^2$ because there are too many vectors.
   • If one vector can be expressed as a linear combination of the others, then the set $S$ is linearly dependent.

Would you like me to test for linear independence of $S$ and provide the solution? Let me know!

Related Questions:

1. What is the span of a set of vectors in $\mathbb{R}^2$?
2. How do we determine if a set of vectors is linearly independent?
3. What conditions must be satisfied for a set to form a basis of $\mathbb{R}^2$?
4. How can we write one vector as a linear combination of others if they are dependent?
5. How does the dimension of a vector space affect the maximum number of linearly independent vectors?

Tip: To test linear independence, construct a matrix with the vectors as columns (or rows) and calculate its rank or determinant!