Whats the answer for these Basis and dimension questions :

Question

Q1:Explain why S is not a basis for R² :

1.S={(-4 , 5),(0 , 0)}

2.S={(-3,2)}

Q2:Explain why S is not a basis for R³:

3.S={(1,3,0),(4,1,2),(-2,5,-2)}

4.S={(1,1,2),(0,2,1)}

Answer 1

(1) You need 2 linearly independent vectors in R² to have a basis for R². The 2 given vectors are not linearly independent.

(2) You need 2 linearly independent vectors in R² to have a basis for R². You have only one vector in this set.

(3) You need 3 linearly independent vectors in R³ to have a basis for R³. The 3 given vectors are not linearly independent. You can verify this by placing the components of the 3 given vectors into the rows of a 3x3 matrix. Evaluate the determinant and see that it is zero.

(4) You need 3 linearly independent vectors in R³ to have a basis for R³. You have only 2 vectors in this set.