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Aim

Theory

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Procedure

Simulation

Posttest

References

Contributors

Feedback

Linear Independence, Basis, Dimension and Co-ordinates

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1. Consider the vector space R² over R. Then (2, 7) can be written as a linear combination of elements of which of the following sets?

a: S₁={(1, 3), (2, 6)} Explanation

Explanation

b: S₂={(1, 0), (0, 0), (2, 0))} Explanation

Explanation

c: S₃={(0, 1), (1, 2), (1, 1)} Explanation

Explanation

d: S₄={(1, 2), (0, 0), (2, 0))} Explanation

Explanation

2. Consider the vector space P₂(x) over R.Then which of the following sets can span P₂(x)?

a: S={1, 2x, x₂+2} Explanation

Explanation

b: S={x-x², -x, x²} Explanation

Explanation

c: S={1, 0, x²-4} Explanation

Explanation

d: S={1, -5x} Explanation

Explanation

3. Consider the vector space R³ over R.Then which of the following sets can span R³?

a: S={(2, 0, -2), (3, 1, 1), (2, 0, 0) (0, 0, 1)} Explanation

Explanation

b: S={(1, 0, 2), (1, -1, 1), (0, 0, 0)} Explanation

Explanation

c: S={(-1, 0, 1), (1, 0, 0) (0, 0, 1)} Explanation

Explanation

d: S={(1, 0, 0), (2, 1, 0) (0, 1, 0)} Explanation

Explanation

4. Consider the vector space R² over R. Then which of the following sets cannot span R²?

a: S={(1, 4), (2, -8)} Explanation

Explanation

b: S={(1, 2), (0, 0), (3, 6)} Explanation

Explanation

c: S={(-3, 4), (6, 7)} Explanation

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d: S={(2, 5), (0, 0)} Explanation

Explanation

5. Consider the vector space R² over R. Then a non-trivial linear combination of elements of which of the following sets equals zero?

a: S={(-2, 3), (1, 4)} Explanation

Explanation

b: S={(-2, 0), (0, 4), (0, 0)} Explanation

Explanation

c: S={(3, 1), (6, 2)} Explanation

Explanation

d: S={(2, 2), (6, 6)} Explanation

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