MATH3200, Lecture 15: Linear Combinations. Bees, Chemical Reaction Systems, and Consistency

Transcription

1 Lecture 15: Linear Combinations: Bees, Chemical Reaction Systems, and Consistency Winfried Just Department of Mathematics, Ohio University February 14, 2018

2 Linear combinations: The definition Definition A vector w is a linear combination of vectors v 1, v 2,..., v n if there exist scalars d 1, d 2,..., d n such that w = d 1 v 1 + d 2 v d n v n. The vectors in the above definition must all be of the same order to make addition meaningful, but they could be either row vectors (of order 1 m) or column vectors (of order m 1). Their common dimension m can be smaller than, equal to, or larger than n. The zero vector 0 of a given order is always a linear combination of any nonempty set of vectors { v 1, v 2,..., v n } of the same order, as we can choose the coefficients d j in the above definition to be all 0.

3 An example of linear combinations: A bee Assume a bee colony nests in a hollow tree. Let [0, 0, 0] denote the position of the hive. A foraging bee travels t 1 = 3 time units in the direction of vector v 1 = [2, 3, 0.5] and then t 2 = 2 time units in the direction of vector v 2 = [0, 2, 0.5] and discovers some tasty blossoms on another tree. We can express the position of these blossoms as the linear combination w = t 1 v 1 + t 2 v 2 = 3[2, 3, 0.5] + 2[0, 2, 0.5] = [6, 5, 0.5]. On the way back to the hive, our foraging bee discovers a better route to the blossoms, and upon return to the hive, she shares this info with her sisters by performing a wiggle dance.

4 Two sisters After the dance, sister Beezee of the foraging bee travels first t 3 time units in the direction of vector v 3 = [1, 1, 0.2] and then t 4 time units in the direction of vector v 4 = [2, 1, 0.3], while sister Buzzy travels first t 5 time units in the direction of vector v 5 = [3, 3, 0.2] and then t 6 time units in the direction of vector v 6 = [0, 1, 0.1]. Homework 38: (a) One of the two sisters didn t pay attention during the wiggle dance and never found the blossoms. Which one was it, and how can you tell? (b) How long did it take the other sister to arrive at the blossoms?

5 Another example: A chemical reaction network Assume a vat contains a mixture of chemical compounds (or, more generally, species) A, B, C, D. Chemists know that the following reactions could in principle occur: A + 2B 2C A + 2C 2D A + B D B + D 2C Homework 39: If the compounds in this system were oxygen O 2, carbon monoxide CO, carbon dioxide CO 2 and carbon C, how would these match to the letter symbols A, B, C, D above? Which directions of the above reactions would be ergetically implausible under normal conditions? We will from now on just work with letters A, B, C, D that could represent various chemical species and assume that each reaction could proceed in either direction.

6 Hint for Homework 39 If the chemical species in this system are oxygen O 2, carbon monoxide CO, carbon dioxide CO 2 and carbon C, the only possible reactions are O 2 + C CO 2 O 2 + 2C 2CO O 2 + 2CO 2CO 2 C + CO 2 2CO

7 Acknowledgement This example is loosely adapted from Gerhard Just et al. (1988) Mathematik für Chemiker. (Mathematics for Chemists.) 3rd edition. VEB Deutscher Verlag für Grundstoffindustrie. It was developed by our family business in discussions with the first author of this book.

8 A chemical reaction network, continued Chemists know that the following reactions could in principle occur: A + 2B 2C A + 2C 2D A + B D B + D 2C However, they do not know which of these reactions actually occur and want to figure it out by observing how the the concentrations change over time. The initial concentrations (in moles per volume of the vat) will be denoted by [A] 0, [B] 0, [C] 0, [D] 0. Now suppose concentrations are measured again after one hour and denoted by [A] 1, [B] 1, [C] 1, [D] 1. Let w = [[A] 1 [A] 0, [B] 1 [B] 0, [C] 1 [C] 0, [D] 1 [D] 0 ]. The vector w represents the net change in concentrations. If some coordinate [X ] 1 [X ] 0 is positive, then a net production of compound X was observed, if some coordinate [X ] 1 [X ] 0 is negative, then a net consumption of compound X was observed.

9 Which reactions do occur? A + 2B 2C A + 2C 2D A + B D B + D 2C Chemists know that only the above reactions could occur, but suspect that not all of them in fact do. (When) can one deduce, based exclusively on observing the vector w of net change of concentrations, that at least one, at least two, or at least three of these reactions do occur? (When) can we further narrow down the possibilities based on energetic constraints that rule out one direction for each of the four reactions?

10 Vectors for each reaction 1 A + 2B 2C 2 A + 2C 2D 3 A + B D 4 B + D 2C Consider the vector v 1 = [ 1, 2, 2, 0]. This vector represents the net change in concentrations if only the first reaction occurs and consumes one mole of the first reactant, which is compound (or species) A in this case. Homework 40: (a) Find analogues v 2, v 3, v 4 of the vector v 1 for reactions 2, 3, and 4. (b) How is the vector w = [[A] 1 [A] 0, [B] 1 [B] 0, [C] 1 [C] 0, [D] 1 [D] 0 ] related to the vectors v 1, v 2, v 3, v 4?

11 Systems of linear equations as linear combinations Consider a system of linear equations a 11 x 1 + a 12 x a 1n x n = b 1... a m1 x 1 + a m2 x a mn x n = b m Let a 1,..., a n be the column vectors of the coefficient matrix A, and let b be the column vector that represents the right-hand side: a 1 = a 11. a m1 a 2 = a 12. a m2... a n = Then the above system can be written as x 1 a 1 + x 2 a x n a n = b. a 1n. a mn b 1 b =. b m

12 Systems of linear equations as linear combinations Consider a system of linear equations 6x 1 x 2 + x 3 = 5 x 2 7x 3 = 0 x 1 + 2x 2 3x 3 = 9 Homework 41: Find vectors a 1,..., a n such that the above system can be written in the form x 1 a 1 + x 2 a x n a n = b.

13 Consistency of systems and linear combinations a 11 x 1 + a 12 x a 1n x n = b 1... a m1 x 1 + a m2 x a mn x n = b m Let a 1,..., a n be the column vectors of the coefficient matrix A, and let b be the column vector that represents the right-hand side. By writing the system as x 1 a 1 + x 2 a x n a n = b we essentially get a one-line proof of the following result: Theorem In the notation introduced above, the system of linear equations is consistent if, and only if, the vector b is a linear combination of the vecors a 1, a 2,..., a n.

14 Solutions of systems and linear combinations a 11 x 1 + a 12 x a 1n x n = b 1... a m1 x 1 + a m2 x a mn x n = b m Let a 1,..., a n, b be as on the previous three slides. By writing the system as x 1 a 1 + x 2 a x n a n = b, we also see that solving the system is the same as finding all vectors of coefficients x 1, x 2,..., x n that work for the linear combination. Homework 42: Reread the story of Marilyn and Marvin and convince yourself that Marilyn had known this all along.