Practice Linear Independence (23) - Linear Independence - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
Determine if the following vectors are linearly independent: [1, 2], [2, 3].
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Try forming a linear equation with both vectors.
Question 2.
Are the vectors [1, 0] and [0, 1] linearly independent?
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Think about spanning a plane.
3Question 3.
A set of vectors is linearly independent if:
- At least one vector can be expressed as a linear combination of others
- The only solution to their linear combination equaling zero is non-trivial
- The only solution to their linear combination equaling zero is trivial
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Recall what a trivial solution is.
4Question 4.
If a set of vectors are collinear in R^2, are they linearly independent?
- True
- False
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Think about spanning a plane.
Question 5.
Show that the vectors [1, 0, 0], [0, 1, 0], [0, 0, 1] form a basis for R^3.
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Check for linear combinations.
Question 6.
Given vectors in R^3, [x, 2x, 3x], under what conditions for 'x' are they dependent?
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Consider the implications of scalar multiples.

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