Winter 2026
MATH 504: Abstract linear algebra
Branko Ćurgus

Tuesday, January 6, 2026

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• My notes on Vector Spaces.

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Monday, January 5, 2026

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• Professor Sheldon Axler, the author of Linear Algebra Done Right has made PDF file of the fourth edition of this popular linear algebra book available free of charge. We are taking advantage of this generous gesture and we are using this book this quarter.
• The information sheet
• Some relevant Wikipedia links:
  • Set
  • Catesian product
  • Function
  • Commutative group, also called Abelian group
  • Field
  • Real numbers
  • Complex numbers
  • Vector space
  • Pointwise operations among functions
• Before briefly reflecting on the history of Linear Algebra, I want to celebrate the history of mathematics with a short paragraph:
  Throughout history, human civilizations have developed and shared mathematical knowledge. Successive civilizations have recognized and admired the contributions of those who came before. Earlier discoveries have provided both a foundation and an inspiration for further advances, giving mathematics the spirit of a collective endeavor of humanity across the past, the present, and the future.
• What is the oldest linear algebra problem?
  • Clay tablet VAT 8389 from the Old Babylonian period, from 2000 to 1600 BC, contains what is believed to be the earliest word problem that can be interpreted as a system of linear equations:
    Total area of two fields is 1800 sar, the rent for one is 2 silà of grain per 3 sar, for the other is 1 silà per 2 sar, the total rent on the first exceeds that on the other by by 500 silà. What is the area of each plot?
    This blog has the picture of clay tablet VAT 8389 and more details about it.
    
    A translation of this word problem into a system of linear equations is as follows: \begin{alignat*}{4} &x_1 & &\ + &x_2 & = 1800 \\ \tfrac{2}{3} &x_1 & &- \tfrac{1}{2} &x_2 & = \phantom{1}500. \end{alignat*}
  • Problem 40 of the Rhind papyrus which is dated to 1550 BC is:
    Divide 100 hekats of barley among 5 men so that the common difference is the same and so that the sum of the two smallest is 1/7 the sum of the three largest.
    Since the Rhind papyrus was copied by the scribe Ahmes from a now-lost text from the period around 1850 BC, and this lost text might have been copied from an even older text from around 2500 BC, the above problem could be by far the oldest known linear algebra problem.
    
    Denote by $x_1$ the smallest number and by $x_2$ the common difference. After simplification the above problem translates into the following system of linear equations: \begin{alignat*}{5} 5 &x_1 & & + 10 &x_2 & = 100 \\ \tfrac{11}{7} &x_1 & & - \phantom{1}\tfrac{2}{7} &x_2 & = \phantom{10}0. \end{alignat*}
  • Most importantly for us, the oldest known treatment of systems of linear equations from antiquity which resembles the methods that we will use in this class is in Chapter 8 of the Chinese textbook Nine Chapters of the Mathematical Art which is at least 1800 years old.
    From 3 top-grade rice paddies, 2 medium-grade, and 1 low-grade, the combined yield is 39 dou of grain. From 2 top-grade, 3 medium-grade, and 1 low-grade, the combined yield is 34 dou of grain. From 1 top-grade, 2 medium-grade, and 3 low-grade, the combined yield is 26 dou of grain. How much dou does one bundle of each grade yield?
    Denote by $x_1$ the yield of the top-grade rice paddy, by $x_2$ the yield of the medium-grade, and by $x_3$ the yield of the low-grade rice paddy. Then the above problem translates into the following system of linear equations: \begin{alignat*}{7} 3 &x_1 & & + 2 &x_2 & + & x_3 = 39 \\ 2 &x_1 & & + 3 &x_2 & + & x_3 = 34 \\ &x_1 & & + 2 &x_2 & + 3 & x_3 = 26 \end{alignat*}
• If the history of mathematics might inspire you to study mathematics with more enthusiasm, below I link to some websites with more about the history of Linear Algebra.
  • Early History of Linear Algebra by Roger Hart
  • History of matrices
  • History of abstract vector spaces
  • Solving a System of Linear Equations Using Ancient Chinese Methods by Mary Flagg

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