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We see that the matrices (of a given size m x n) with components in a field K form a vector space over K which we may denote by MatmxnCK). We define one more notion related to a matrix. Let A = (ai) be an m x n matrix. The n x m matrix B = (b j ;) such that bji = aij is ealled the transpose of A, and is also denoted by t A. Taking the transpose of a matrix amounts to ehanging rows into eolumns and vice versa. If A is the matrix whieh we wrote down at the beginning of this seetion, then tA is the matrix a 12 a 21 a 22 a 31 a 32 a 1n a 2n a 3n C ...

Example 5. Let V, V' be any vector spaces over the field K. The mapping which associates the element 0 in V' to any element u of V is called the zero mapping and is obviously linear. It is also denoted by o. 54 [III, §2] LINEAR MAPPINGS As an exercise (Exercise 2) prove: Let L: V --+ W be a linear map. Then L(O) = O. In particular, if F: V --+ W is a mapping and F(O) # 0 then F is not linear. Example 6. Tbe space of linear maps. Let V, V' be two vector spaces over the field K. We consider the set of all linear mappings from V into V', and denote this set by 2'(V, V'), or simply 2' if the reference to V, V' is clear.

Vr } is a basis of W, then we let U be the space generated by {v r + l' ... ,vn }· We note that given the subspace W, there exist usually many subspaces U such that V is the direct sum of Wand U. ) In the section when we discuss orthogonality later in this book, we shall use orthogonality to determine such a subspace. 3. •• ,ws} a basis of W Every element of U has a unique expression as a linear combination X1U 1 + ... + XrU r' with XiEK, and every element of W has a unique expression as a linear combination Y1 W 1 + ...