Determine whether the set of vectors is a basis for R3 Deter

Determine whether the set of vectors is a basis for R^3.

Determine whether the set of vectors is a basis for R^3. 2) Given the set of vectors {[1 0 0],[0 1 0],[0 0 1],[0 1 1]}, decide which of the following statements is true: A: Set is linearly independent and spans R^3. Set is a basis for R^3. B: Set is linearly independent but does not span R^3. Set is not a basis for R^3. C: Set spans R^3 but is not linearly independent. Set is not a basis for R^3. D:Set is not linearly independent and does not span R^3. Set is not a basis for R^3.

Solution

2.The 1st, 2nd and the 3rd vectors are linearly independent. The 4th vector is the sum of the 1st and the 2nd vectors. Further, any vector in R³ is a linear combination of the first 3 vectors. Thus, { ( 1,0,0)^T, ( 0, 1,0)T , ( 0,0,1)T} form a basis for R³ .The statement C is true. The given set spans R^{3 ,} but it is not linearly independent. Therefore, it is not a basis for R³.