Tell whether or not set S = {(3, 4, 6), (4, 9, 1)} spans R³

The set \( S = \{(3, 4, 6), (4, 9, 1)\} \) does **not** span \( \mathbb{R}^3 \).

### Reason:
In order for a set of vectors to span \( \mathbb{R}^3 \), the set must contain at least 3 linearly independent vectors because \( \mathbb{R}^3 \) is a 3-dimensional space. The set \( S \) contains only 2 vectors, which is insufficient to span \( \mathbb{R}^3 \).

More formally, for two vectors to span \( \mathbb{R}^3 \), there would need to be a third linearly independent vector added to the set. Since we only have two vectors, the span of these vectors will at most cover a plane within \( \mathbb{R}^3 \), not the entire space.

Would you like to explore the concept of spanning, or need further clarification?

Here are 5 related questions to expand this topic:
1. What does it mean for a set of vectors to span a space?
2. How can you check if a set of vectors is linearly independent?
3. Can two vectors ever span \( \mathbb{R}^3 \), and under what conditions?
4. What is the role of the dimension of a space in determining spanning sets?
5. How can you construct a set that spans \( \mathbb{R}^3 \)?

**Tip:** A set of vectors spans a space if any vector in that space can be written as a linear combination of the vectors in the set.

Explore whether the set of vectors S = {(3, 4, 6), (4, 9, 1)} spans the 3-dimensional space R³. This problem involves key linear algebra concepts such as vector spaces, linear independence, and the spanning set theorem. Suitable for college-level students.

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