26.4. Linear Combination and Span

Try these first

1. 1.

   What is a linear combination of the vectors (2, 3) and (1, -1)?

   Hint

   Try substituting different values for a and b.

2. 2.

   Define the span of the set S = {(1, 2), (2, 3)}.

   Hint

   Think about how many combinations can be formed.

3. 3.

   What is a linear combination of two vectors?

   • A vector formed by adding two vectors
   • A vector formed by scaling and adding two vectors
   • A vector formed only from one of the vectors

   Hint

   Think about how we create new vectors from existing ones.

4. 4.

   The span of a set of vectors is always:

   • True
   • False

   Hint

   Remember the definition of span and its properties.

5. 5.

   Let v₁ = (2, 3) and v₂ = (4, 6). Demonstrate whether these vectors are linearly independent or dependent by showing a linear combination.

   Hint

   Use the definition to write out the coefficients that lead to 0.

6. 6.

   Given a set of vectors {v₁, v₂, v₃} in R³, where v₁ = (1, 0, 0), v₂ = (0, 1, 0), v₃ = (0, 0, 1). Explain the significance of their span.

   Hint

   Think about how the bases span the entire axis.