A set of two three or four red vectors in R2 or R3 is shown

A set of two, three, or four red vectors in R^2 or R^3 is shown in each picture below. Determine whether each set of vectors is linearly independent or linearly dependent.

Solution

Result:  We know that, in a vector space V if dimension of V is n,then any set of (n+1) vectors is linearly dependent.

1. Dimension of R² is 2.Given a set of 3-vectors.Therefore they are Linearly dependent.

2. Given two vectors are perpendicular. Therefore they are Linearly independent.

3. Dimension of R³ is 3.Given a set of 4-vectors.Therefore they are Linearly dependent.

4. Two vectors are in same plane with different direction.So they are Linearly independent.The third vector is in a different plane.So this vector is Linearly independent qith previous two vectors. Therefore all the 3 vectors are Linearly independent.

5. Dimension of a plane is 2.Therefore given 3 vectors in a same plane are Linearly dependent.

6. given two vectors are in different plane.Therefore they are Linearly independent.