24.4. Linear Combination and Span

Try these first

1. 1.

   Write a linear combination of the vectors (1, 2) and (3, 4).

   Hint

   Use coefficients of your choice.

2. 2.

   What is the span of the vectors (1, 0) and (0, 1)?

   Hint

   Think about what combinations can result from these vectors.

3. 3.

   What is a linear combination?

   Hint

   Consider expressions of the form a1v1 + a2v2.

4. 4.

   True or False: The span of vectors is the collection of all linear combinations of those vectors.

   • True
   • False

   Hint

   Remember definitions covered earlier.

5. 5.

   Given the vectors v1 = (1, 1) and v2 = (2, 2), find a unique combination resulting in the vector (3, 3). Discuss whether these two vectors are independent.

   Hint

   Draw or graph these vectors to visually speculate about their relationship.

6. 6.

   Design a three-dimensional span using two specific non-parallel vectors. Discuss if it can fill the entire space.

   Hint

   Visualize using a 3D graph to see how dimensions interact!