8.1 Which of the following statements about linear vector spaces are true? Where a statement is false, give a counter-example to demonstrate this. (a) Non-singular N x N matrices form a vector space...

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Extracted text: 8.1 Which of the following statements about linear vector spaces are true? Where a statement is false, give a counter-example to demonstrate this. (a) Non-singular N x N matrices form a vector space of dimension N2. Singular N x N matrices form a vector space of dimension N2. Complex numbers form a vector space of dimension 2. (d) Polynomial functions of (e) Series fao, a1, a2, . . . , aN for which la,= 1 x form an infinite-dimensional vector space. form an N-dimensional vector space. Absolutely convergent series form an infinite-dimensional vector space. Convergent series with terms of alternating sign form an infinite-dimensional vector space.