answer all questions or I will give you a thumbs down. double check your answers

Question

answer all questions or I will give you a thumbs down. double check your answers.

2. State Truc" or alsc" for cach he owing statements, You do not need to justify your answers (a) There cxists 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 spacc 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 can be expanded by four vec- ors ectr (h) Any vector space of dimension can be expanded by two vec- (i) Three vectors are linearly dependent if and only if one of G) The column space and row space of the same matrix A will tors. them can be written as a linear combination of the other two have the same dimension

Explanation / Answer

a) There exists a vector space consisting of exactly 100 vectors:

False.

b) There exists a vector space of dimension 100:

True.

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

True.

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 other two:

True.

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

True.

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