True Or False???

Question

a) If four vectors span R 4 ; then they must form a basis of R 4 ......

b) Four vectors in R 3 must be linearly independent......

c) Two nonzero vectors are linearly independent if one is not scalar multiple of the other......

d) The zero vector is linearly independent......

Answer 1

Linear algebra has some connected ideas that come up in parts a and b. The number of dimensions in a space (such as 4 in R₄) equals the number of vectors in the basis. The basis will always span the space it represents and will have linearly independent vectors.

a) So that is why part a is true. The basis will span the set in question (plus it will have linearly independent vectors).

b) Part b would be impossible because R₃ will have a basis of 3 linearly independent vectors. If you throw in a 4th vector, it will be related to the other 3 (called a linear combination of the other 3) which is an informal way of saying the set of 4 vectors will be linearly dependent.

c) Two vectors are linearly independent unless they point in the same or opposite directions (which would put them on the same line and make them linearly dependent). If one nonzero vector is a multiple of the other then they do point in the same or opposite direction such as 2i + j and 6i + 3j. So your statement c is true.

d) The zero vector v is linearly dependent because if you take the equation cv = 0 with scalar c where v is the zero vector, then there is a solution to the equation where c is not 0. In fact, c can equal any real number! (If v was linearly independent, the only solution to cv = 0 would be c = 0.) This means d is false.