Virtual Labs

Tools

Aim

Theory

Pretest

Procedure

Simulation

Posttest

References

Contributors

Feedback

Aim

Theory

Pretest

Procedure

Simulation

Posttest

References

Contributors

Feedback

Linear Independence, Basis, Dimension and Co-ordinates

Choose difficulty:

Beginner

Intermediate

Advanced

1. Consider the vector space R² over R. Which of the following sets is linearly independent?

a: S₁={(3, 9), (1, 3)} Explanation

Explanation

b: S₂={(1, 2), (0, 0)} Explanation

Explanation

c: S₃={(2, 7), (1, 4), (1, 0)} Explanation

Explanation

d: S₄={(3, 6), (1, 5)} Explanation

Explanation

2. Consider the vector space R² over R. Which of the following sets is linearly dependent?

a: S₁={(2, 6), (0, 3)} Explanation

Explanation

b: S₂={(1, 0), (0, 1)} Explanation

Explanation

c: S₃={(3, 4), (4, 9)} Explanation

Explanation

d: S₄={(1, 2), (2, 4)} Explanation

Explanation

3. Consider the vector space R³ over R. Which of the following is its basis?

a: B₁={(1, 0, 0), (0, 2, 2), (0, 0, 0)} Explanation

Explanation

b: B₂={(1, 0, 0), (0, 1, 3), (0, 2, 4)} Explanation

Explanation

c: B₃={(2, 3, 0), (1, 1, 0), (1, 0, 0)} Explanation

Explanation

d: B₄={(1, 0, 2), (1, 0, 4)} Explanation

Explanation

4. Consider the vector space P₂(x) over R. Which of the following is its basis?

a: B₁={1, 0, x₂} Explanation

Explanation

b: B₂={2, x+2, x²+4} Explanation

Explanation

c: B₃={1+x, 1+x²} Explanation

Explanation

d: B₄={2+x, -x, 2} Explanation

Explanation

5. Consider the vector space R² over R. Which of the following is true?

a: Dim of R² > 2, dim ϕ = 1 Explanation

Explanation

b: Dim of R² < 2, dim ϕ = 0 Explanation

Explanation

c: Dim of R² = 2, dim ϕ = 0 Explanation

Explanation

d: Dim of R² ≠ 2, dim ϕ = 1 Explanation

Explanation

6. Consider the vector space R² over R. Co-ordinates of (1, 2)∈R² w.r.t. the basis {(0, 1) , (-1, 0)} are:

a: 1 & 2 Explanation

Explanation

b: -1 & 2 Explanation

Explanation

c: -2 & 1 Explanation

Explanation

d: 2 & -1 Explanation

Explanation

Community Links Sakshat Portal Outreach Portal FAQ: Virtual Labs

AGPL 3.0 & Creative Commons (CC BY-NC-SA 4.0)