By Jørn Børling Olsson

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F/. Let m denote the highest degree of the polynomials in this list. Then every polynomial in the span of this list has degree at most m. Thus z mC1 is not in the span of our list. F/. F/ is inﬁnite-dimensional. v1 ; : : : ; vm /. By the deﬁnition of span, there exist a1 ; : : : ; am 2 F such that v D a1 v1 C C am vm : Consider the question of whether the choice of scalars in the equation above is unique. v1 ; : : : ; vm /. If the only way to do this is the obvious way (using 0 for all scalars), then each aj cj equals 0, which means that each aj equals cj (and thus the choice of scalars was indeed unique).

In general, a vector space is an abstract entity whose elements might be lists, functions, or weird objects. B Deﬁnition of Vector Space 15 Soon we will see further examples of vector spaces, but ﬁrst we need to develop some of the elementary properties of vector spaces. The deﬁnition of a vector space requires that it have an additive identity. The result below states that this identity is unique. 25 Unique additive identity A vector space has a unique additive identity. Proof Suppose 0 and 00 are both additive identities for some vector space V.

To show that U \ W D f0g, suppose v 2 U \ W. Then there exist scalars a1 ; : : : ; am ; b1 ; : : : ; bn 2 F such that v D a1 u1 C C am um D b1 w1 C C bn wn : Thus a1 u1 C C am u m b1 w1 bn wn D 0: Because u1 ; : : : ; um ; w1 ; : : : ; wn is linearly independent, this implies that a1 D D am D b 1 D D bn D 0. Thus v D 0, completing the proof that U \ W D f0g. B 1 Find all vector spaces that have exactly one basis. 28. x1 ; x2 ; x3 ; x4 ; x5 / 2 R5 W x1 D 3x2 and x3 D 7x4 g: Find a basis of U. 4 (b) Extend the basis in part (a) to a basis of R5 .