By C. E. Cullis

Initially released in 1918, this publication varieties a part of a three-volume paintings created to extend upon the content material of a sequence of lectures added on the collage of Calcutta in the course of the iciness of 1909-10. the manager characteristic of all 3 volumes is they care for oblong matrices and determinoids as unique from sq. matrices and determinants, the determinoid of an oblong matrix being on the topic of it within the similar method as a determinant is expounded to a sq. matrix. An try out is made to set forth a whole and constant concept or calculus of oblong matrices and determinoids. the second one quantity includes additional advancements of the final thought, together with a dialogue of matrix equations of the second one measure. It additionally incorporates a huge variety of purposes to algebra and to analytical geometry of house of 2, 3 and n dimensions.

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76 Let A 6 Mn(F) and write A = (o-i, a2, . . , an), where each on e F™. (a) Show that dim(Span{ai,a2, • . • ,an}} = r(A). (b) Let P be an n x n invertible matrix. , Pan] = (fa, fc, . . , /3n). ir are linearly independent if and only if /^ , /3j2 , . . , Pir are linearly independent (over F). 77 Let W\ and W2 be the vector spaces over M spanned, respectively, by and Find the dimensions and bases for W\ fl W? and W\ + W?. 78 Let a,ij be integers, 1 < i, j ' < n. If for any set of integers bi, b%, .

1) {(x,y) | y — 3x, x, y e R} is a subspace of E2. (m) {(x, y) | x — y = 1, x, y e E } is a subspace of R2. 14 Consider Pn [x] and P[x] over E. Answer true or false: (a) {p(x) | p(x) = ax + b, a, b € E} is a subspace of PS[X]. (b) {p(x) | p(x) = ax2, a 6 R} is a subspace of PS[X]. (c) {p(x) | p(x) = a + x 2 , a e R } i s a subspace of PS[X]. (d) (p(x) | p(x) € P[x] has degree 3 } is a subspace of P[x]. (e) {p(x) | p(0) = 0, p(x) e P[x] } is a subspace of P[x]. (f) {p(x) | p(0) = 1, p(x) G P[x] } is a subspace of P[x].

An, f3 are linearly dependent, show that /3 can be uniquely expressed as a linear combination of ai, 0:2, . ,an. 30 Show that the vectors ai(^ 0),a2, . . ,a n of a vector space V are linearly dependent if and only if there exists an integer k, 1 < k < n, such that afe is a linear combination of ai, a 2 , . . 31 Let V and W be vector spaces over F. Denote by V x W the collection of all ordered pairs (v, w), where v &V and w e W, and define (Vi, Wi) + (V2, W2) = («1 + U2> Wi + W2) and fc(v, ty) = (kv, kw), k e F.

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