Practice Algebraic Criterion For Linear Independence (23.5)

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6 sample questions

Question 1.
Check if the vectors [1, 2], [2, 4] are linearly independent.
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Look for scalar multiples.
Question 2.
Are the vectors [1, 1] and [2, 2] independent?
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Can you express one in terms of the other?
3Question 3.
What defines linear independence?
- Only trivial solutions exist for linear combinations.
- At least one non-trivial solution exists.
- Vectors are multiples of each other.
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Think about the solution types.
4Question 4.
True or False: A homogeneous system can have non-trivial solutions.
- True
- False
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Recall the definitions of trivial and non-trivial solutions.
Question 5.
Consider the vectors v_1 = [1, 2, 1], v_2 = [2, 4, 2], and v_3 = [4, 8, 4]. Show that these vectors are linearly dependent.
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Try row-reduction to explore relationships.
Question 6.
For the vectors v_1 = [1, 2, 3] and v_2 = [1, a, 3], determine the value(s) of 'a' for which they are independent.
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Set up your system and check the determinant.

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