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19 February, 10:25

Which of the following statements is false?

a. R3 is a vector space

b. P2 is a vector space

C. M2x2 is a vector space

d. The set of all polynomials of degree 4 is a vector space

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  1. 19 February, 10:36
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    C. M2x2 is a vector space

    Step-by-step explanation:

    According to the definition, the each element in a vector spaces is a vector. So, 2*2 matrix cannot be element in a vector space since it is not even a vector.

    The set of all ordered triples of real numbers is called 3-space, denoted R 3 ("R three"). See Figure ... Vectors in R 3 are called 3-vectors (because there are 3 components), and the geometric descriptions of addition and scalar multiplication given for 2-vectors also carry over to 3-vectors.

    Since every polynomial of degree up to 2 is also a polynomial of degree up to 3, P2 is a subset of P3. And we already know that P2 is a vector space, so it is a subspace of P3.

    With addition, the set of polynomials of degree 2 almost form a vector space, but there are some problem. The first one, is that the zero vector, i. e. the zero polynomial is not of degree 2 ... If you really need them to form a vector space, then you should consider the set of all polynomials of degree at most 2.
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