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1 December, 15:09

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.

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  1. 1 December, 15:19
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    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.
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