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3 cards from this section. Quick, good the night before a test.
Determine whether the vectors v₁ = [1] and v₂ = [2] are linearly independent.
Think about whether one can be written as a multiple of the other.
What is meant by 'linear combination'?
Look at the format: a₁*v₁ + a₂*v₂ + ... + aₙ*vₙ.
What is a linear combination?
Remember its definition involving scalar multiplication.
True or False: If a set of vectors is linearly independent, at least one vector can be expressed as a combination of the others.
Think about the implication of independence.
Given three vectors, v₁ = [1, 2, 3], v₂ = [2, 4, 6], and v₃ = [3, 6, 9], show they are linearly dependent using row reduction.
Check for redundancy in your elimination steps.
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.
Set the coefficients of the linear combination equal to zero and solve.
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