In IB we introduced the characteristic functions c S for subsets S of a set U and proved (IB2) th... more In IB we introduced the characteristic functions c S for subsets S of a set U and proved (IB2) that the function c S ↦ S is a bijection between K U and P(U). Subsequently it was to be verified (Exercise IB3) that this same bijection made the assignments c S + c T ↦ S + T and c S c T ↦ S ∩ T. We have thereby that (P(U), +, ∩) is “algebra-isomorphic” to the commutative algebra (K U , +, ·), and hence (P(U), +, ∩) is a commutative algebra over the field K. In particular, (P(U), +) is a vector space over K, while (P(U), +, ∩) is a commutative ring; ∅ is the additive identity and U itself is the multiplicative identity. For the present we shall be concerned only with the vector space structure.
... I. Servatius. Brigitte. 1954-. II. Servatius. Herman. 1957-. III. Title. IV. Series. QA166.6.... more ... I. Servatius. Brigitte. 1954-. II. Servatius. Herman. 1957-. III. Title. IV. Series. QA166.6.G73 1993 511'.6-dc20 93-34431 CIP Copying and reprinting. ... including rigid ones such as pedestals orbridges. as well as moving structures such machines or organic molecules. ...
These are the lecture notes for a short course presented at the University of Alberta in Edmonton... more These are the lecture notes for a short course presented at the University of Alberta in Edmonton, Alberta, in March of 1966. The first two thirds of these notes give an introduction to the theory of matroids and is based on two fundamental papers of the subject: [2], On the abstract properties of linear dependence; and [1], Lectures on Matroids. In the last third of these notes, matroid theory is applied to the theory of graphs. Most of the results obtained here were originally worked out by H. Whitney in a series of papers preceding [2]. In fact it is evident that the work that went into these papers led to and finally culminated in his matroid papers.
[1] W. T. Tutte, (1965). Lectures on Matroids, J. Res. Nat. Bur. Stand. 69B, 1–48.
[2] H. Whitney, (1935). On the abstract properties of linear independence, Amer. J. Math. 57, 509–533.
In IB we introduced the characteristic functions c S for subsets S of a set U and proved (IB2) th... more In IB we introduced the characteristic functions c S for subsets S of a set U and proved (IB2) that the function c S ↦ S is a bijection between K U and P(U). Subsequently it was to be verified (Exercise IB3) that this same bijection made the assignments c S + c T ↦ S + T and c S c T ↦ S ∩ T. We have thereby that (P(U), +, ∩) is “algebra-isomorphic” to the commutative algebra (K U , +, ·), and hence (P(U), +, ∩) is a commutative algebra over the field K. In particular, (P(U), +) is a vector space over K, while (P(U), +, ∩) is a commutative ring; ∅ is the additive identity and U itself is the multiplicative identity. For the present we shall be concerned only with the vector space structure.
... I. Servatius. Brigitte. 1954-. II. Servatius. Herman. 1957-. III. Title. IV. Series. QA166.6.... more ... I. Servatius. Brigitte. 1954-. II. Servatius. Herman. 1957-. III. Title. IV. Series. QA166.6.G73 1993 511'.6-dc20 93-34431 CIP Copying and reprinting. ... including rigid ones such as pedestals orbridges. as well as moving structures such machines or organic molecules. ...
These are the lecture notes for a short course presented at the University of Alberta in Edmonton... more These are the lecture notes for a short course presented at the University of Alberta in Edmonton, Alberta, in March of 1966. The first two thirds of these notes give an introduction to the theory of matroids and is based on two fundamental papers of the subject: [2], On the abstract properties of linear dependence; and [1], Lectures on Matroids. In the last third of these notes, matroid theory is applied to the theory of graphs. Most of the results obtained here were originally worked out by H. Whitney in a series of papers preceding [2]. In fact it is evident that the work that went into these papers led to and finally culminated in his matroid papers.
[1] W. T. Tutte, (1965). Lectures on Matroids, J. Res. Nat. Bur. Stand. 69B, 1–48.
[2] H. Whitney, (1935). On the abstract properties of linear independence, Amer. J. Math. 57, 509–533.
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[1] W. T. Tutte, (1965). Lectures on Matroids, J. Res. Nat. Bur. Stand. 69B, 1–48.
[2] H. Whitney, (1935). On the abstract properties of linear independence, Amer. J. Math. 57, 509–533.
[1] W. T. Tutte, (1965). Lectures on Matroids, J. Res. Nat. Bur. Stand. 69B, 1–48.
[2] H. Whitney, (1935). On the abstract properties of linear independence, Amer. J. Math. 57, 509–533.