23.2. Linear Combination of Vectors

Try these first

1. 1.

   Determine whether the vectors v₁ = [1] and v₂ = [2] are linearly independent.

   Hint

   Think about whether one can be written as a multiple of the other.

2. 2.

   What is meant by 'linear combination'?

   Hint

   Look at the format: a₁*v₁ + a₂*v₂ + ... + aₙ*vₙ.

3. 3.

   What is a linear combination?

   • A sum of vector magnitudes
   • A combined vector formed using scalars
   • A geometric shape

   Hint

   Remember its definition involving scalar multiplication.

4. 4.

   True or False: If a set of vectors is linearly independent, at least one vector can be expressed as a combination of the others.

   • True
   • False

   Hint

   Think about the implication of independence.

5. 5.

   Given three vectors, v₁ = [1, 2, 3], v₂ = [2, 4, 6], and v₃ = [3, 6, 9], show they are linearly dependent using row reduction.

   Hint

   Check for redundancy in your elimination steps.

6. 6.

   For the vectors v₁ = [1, 1], v₂ = [a, a] where a is a scalar, find values of a for which the vectors are linearly independent.

   Hint

   Set the coefficients of the linear combination equal to zero and solve.