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**Example text**

I = (a il9 · · · , O and x = (<^, · · · , ξη). One can then say that N consists of the vectors x orthogonal to the rows of A, or of the vectors in $Fn that are orthogonal io the span S(ri9 · · · , rn) of the rows. 53 6. 3. Addition Natural rules for addition and multiplication of matrices of scalars can be derived from the corresponding rules for linear transformations ; they can be extended to certain matrices whose elements are not scalars. Linear transformations from 3Fn to 3Fk can be added ; it follows that we can add any two matrices of elements of SF that have the same number of rows and the same number of columns.

Exactly how does L map D onto R ? We know already that L takes the subspace N 36 1. ALGEBRAIC THEORY OF VECTOR SPACES Fig. 4-2 of Z>, the null space, into the zero vector in R. But what does L do to, say, a complementary subspace N' of N in D (Fig. 4-2) ? Let us consider the restriction L' of L to N'. We will show that the linear transformation L' still has range R, but that it is one to one. First of all suppose z belongs to R ; that is, z = Lx for some vector x in D. We can write x = y + y' with j> in N9 y' in TV' (Sec.

14. Let V be the vector space of all polynomials in t with real or com plex coefficients. Characterize V*. Prove that dim V* is larger than dim F (in the sense of Sec. 5). 50 1. ALGEBRAIC THEORY OF VECTOR SPACES 6. RECTANGULAR MATRICES A linear transformation from 2Fn to ^ f c can be represented by a k x n matrix of scalars. This representation provides the motivation for the customary definitions of sum, product, and rank of matrices. A linear trans formation L from an arbitrary «-dimensional vector space F to a fc-dimensional vector space W over the same field has a variety of associated matrices, namely, one for every choice of ordered bases in V and W.