By Walter E. Thirring

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To emphasize its dependence on the scalars, dimC V and dimR V are also used. A vector space of dimension N is sometimes denoted by VN . 4) set of scalars {α1 , α2 , . . , αn } such that |a = N i=1 αi |ai . components of a vector is called the components of |a in the basis B. 2. • • • • • • • • • The number 1 (or any nonzero real number) is a basis for R, which is therefore one-dimensional. √ The numbers 1 and i = −1 (or any pair of distinct nonzero complex numbers) are basis vectors for the vector space C over R.

In particular, the sum of two polynomials of degree less than or equal to n is also a polynomial of degree less than or equal to n, and multiplying a polynomial with complex coefficients by a complex number gives another polynomial of the same type. Here the zero polynomial is the zero vector. The set Prn [t] of polynomials of degree less than or equal to n with real coefficients is a vector space over the reals, but it is not a vector space over the complex numbers. Let Cn consist of all complex n-tuples such as |a = (α1 , α2 , .

Note that although the vectors are the same as in the preceding item, changing the nature of the scalars changes the dimensionality of the space. The set {ˆex , eˆ y , eˆ z } of the unit vectors in the directions of the three axes forms a basis in space. The space is three-dimensional. A basis of Pc [t] can be formed by the monomials 1, t, t 2 , . . It is clear that this space is infinite-dimensional. A basis of Cn is given by eˆ 1 , eˆ 2 , . . , eˆ n , where eˆ j is an n-tuple that has a 1 at the j th position and zeros everywhere else.