True or false and why?
(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 in? nitely 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 and explanation:

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

False. All vector space consists of an infinite number of vectors.

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

True. There exist vector spaces of N dimensions. You can peak an N integer you desire (in this case N=100)

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

False. Thre different vectors could be linearly dependent. Example: (1,1,0),(1,1,1),(0,0,1). The first and the third vector form the second one.

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

True. The maximum number of linearly independent vectors in a vector space of dimension 3 is 3. Therefore any additional vector will be linearly dependent with the others.

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

False. Any vector space has an infinite number of subspaces, independently of its dimension.

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

True. I explained this in the previous statement.

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

False. You expand subspaces of dimension n by adding m linearly dependent vectors to complete the space of dimension (n+m) in which exist. You don´t expand vector spaces.

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

False. I explained this in the previous statement.

(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. One of the three vectors could be linearly dependent with one of the other 2 two vectors and linearly independent with the other.

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

True. The rank of any matrix is the dimension of the columns or the rows. Been only a single number, columns space and row space have to have the same dimension. This can be explained with the rank-nullity theorem.