Practice Worked Examples (24.16) - Vector Space - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
Is the set W = {(x, y, z) ∈ R3: 2x + y - z = 0} a subspace? Explain.
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Check the zero vector and the closure properties.
Question 2.
What is a basis for W = { (x,y,z): x=0 }?
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You need to express other variables in terms of the free ones.
3Question 3.
What is a necessary condition for a subset W to be a subspace of R3?
- It must include the zero vector.
- It must be closed under addition.
- It must meet both conditions.
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Think about the properties necessary for vector spaces.
4Question 4.
True or False: Every set of linearly independent vectors forms a basis.
- True
- False
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Remember the definition of a basis.
Question 5.
Determine if the set W = {(x, y, z) ∈ R3: 3x - y + 4z = 5} can be a subspace.
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Check the zero vector first.
Question 6.
Find a basis for the span of vectors {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and determine its dimension.
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Use row reduction to assess linear independence.

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