Question

State "True" or "False" for each of the following statements. You do not need to justify your answers. (a) There exists a vector space consisting of exactly 100 vectors. (b) There exists a vector space of dimension 100. (c) In a vector space of dimension 3, any three vectors are linearly independent. (d) In a vector space of dimension 3, any four vectors are linearly dependent. (e) Any vector space of dimension 2 has exactly two subspaces. (f) Any vector space of dimension 2 has infinitely many subspaces. (g) Any vector space of dimension 3 can be expanded by four vectors. (h) Any vector space of dimension 3 can be expanded by two vectors. (i) Three vectors are linearly dependent if and only if one of them can be written as a linear combination of the other two. (j) The column space and row space of the same matrix A will have the same dimension

Answer 1

All the answers above without justification as asked:

(a) There exists a vector space consisting of exactly 100 vectors. True

(b) There exists a vector space of dimension 100. True

(c) In a vector space of dimension 3, any three vectors are linearly independent. False

(d) In a vector space of dimension 3, any four vectors are linearly dependent. True

e) Any vector space of dimension 2 has exactly two subspaces. False

(f) Any vector space of dimension 2 has infinitely many subspaces. True

(g) Any vector space of dimension 3 can be expanded by four vectors. True

(h) Any vector space of dimension 3 can be expanded by two vectors. False

(i) Three vectors are linearly dependent if and only if one of them can be written as a linear combination of the other two. False

(j) The column space and row space of the same matrix A will have the same dimension False