Abstract
The Government can choose the value of some variables, under its control, for instance: (1) — Official Discount rate; (2) — Amount of T-Bills to be issued; (3) — Tax rate to be applied.
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- 1.
The set of real numbers (correspoding to the points of a straight line) is usually denoted with \(\mathbb {R}\). \(x\in \) \(\mathbb {R}\) means that x is a real number.
- 2.
We will use systematically this convention.
- 3.
We are familiar with the = sign between numbers:
$$\begin{aligned} 2+3=5 \end{aligned}$$hopefully recognized by each reader. We will start to use the equal sign between vectors and matrices. In the case of vectors, \(=\) means that all the corresponding components are equal.
- 4.
For many people “sum” and “addition” are synonimous. In Mathematics the operation which combines two objects (in this case: vectors) is called “addition”, while “sum” is the name reserved for the result. If you add eggs, salt and cheese you make an “addition”, the omelette (hopefully) you get is the “sum”.
- 5.
Concretely, this means that, in order to purchase 1€, you have to pay 1.111155 U$.
- 6.
In Linear Algebra it turns out to be frequent the combined use of (poor) numbers with (noble) arrays like vectors and matrices. In this context it is customary to stress the (poor) nature of numbers saying that they are scalar quantities.
- 7.
As before: “Scalar multiplication” is the operation and “product” is the result.
- 8.
“Transgender” because, while on the lhs we have two scalar multiplications (University level), on the rhs we have two different multiplications: the first one \(\left( \alpha \beta \right) \) which is a primary school stuff, while the second one is a (University level) multiplication.
- 9.
Indeed very nice!
- 10.
In order to make the reader flexible with some popular notations, we will use indifferently both “\(\cdot \)” and “\(\times \)” for multiplication. We will adhere to the general convention that the sequence of two factors — say — : ab means \(a\cdot b\) or \(a\times b.\) This convention will be respected also when we will generalize multiplication.
- 11.
For Social Sciences students, it seems important to stress that an arbitrage is nothing but the opportunity to exploit market imperfections. Making arbitrages, i.e.; exploiting incoherent market positions is non-necessarily evil, but simply an opportunity.
- 12.
In the case \(k=2\), we have informally explored above, this means that \( \mathbf {u, v}\) are linearly independent if and only if none of them is a multiple of the other.
- 13.
“iff” is beyond English. Mathematicians contract into “iff” the sequence of words “if and only if”. We will use it sometimes.
- 14.
This double indexation reflects the ubiquitous nature of a matrix that can be seen as a package of stacked row vectors or a set of column vectors, placed side by side. Think of the starting Example 1.3.1. Had we to deal with the element \(a_{123 \text { }}\) of some matrix A, we would like to understand whether it is the third entry in the row # 12 or the 23rd in the row # 1. In the case that this ambiguity turns out to be relevant, the two (boring, but more refined) notation items are advisable: \(a_{12,3}\) or \(a_{1,23}\).
- 15.
This convention does not hold for Germans, they do prefer E, the initial letter of “Einheit” = unity.
- 16.
“LESS” stands for “Least Exciting Sub Section”.
- 17.
“Transgender” because, while on the lhs we have two scalar multiplications (University stuff), on the rhs we have two different multiplications: the first one \(\left( \alpha \beta \right) \) which is a primary school stuff, while the second one is a scalar multiplication.
- 18.
Commutativity is a piece of what we learned when we were kids:
-
\(a+b=b+a\);
-
\(a\cdot b=b\cdot a\).
When we experienced some clothing, or even before, we became rationally conscious of non-commutativity:
$$\begin{aligned} \left\{ \begin{array}{c@{\quad }c@{\quad }c} \text {underwear * gown } &{} \ne &{} \text {gown * underwear * } \\ &{} &{} \\ \text {underwear * trousers } &{} \ne &{} \text {trousers * underwear * } \end{array} \right. \end{aligned}$$ -
- 19.
Between numbers x, y, their product xy is 0 if and only if at least one of the factors is 0. For matrices the “only” part of the statement does not hold. E.g.:
$$\begin{aligned} \left[ \begin{array}{c@{\quad }c} 1 &{} 1 \\ 1 &{} 1 \end{array} \right] \left[ \begin{array}{r@{\quad }r} 1 &{} 1 \\ -1 &{} -1 \end{array} \right] =\left[ \begin{array}{c@{\quad }c} 0 &{} 0 \\ 0 &{} 0 \end{array} \right] \end{aligned}$$even if no factor is a null matrix.
- 20.
In fact, if \(\alpha \ne 0\) and — say — \(a\ne 0\), it should sound strange that \(\alpha a=0\). The same argument holds for b.
- 21.
Some people could think: “You’re silly because \( x\times 5=5\times x\)”, but as we’re working with matrices, where commutativity is not guaranteed, the precaution turns out to be justified.
- 22.
It’s possible, but there is no need for the reader to learn it.
- 23.
Please pay attention to language. Notions like “piece of surface” and its extension (area) are frequently interchanged. Sentences like “In the San Diego area there is massive illegal immigration of people from Mexico to the USA” are accepted, even if instead of “area” a mathematician would use another word.
- 24.
If we move along the boundaries of this parallelogram, starting from the first side (the one from \(\left[ \begin{array}{c} 0 \\ 0 \end{array} \right] \) to \(\left[ \begin{array}{c} 1 \\ 3 \end{array} \right] \) and so on, the parallelogram always stays on our right: this explains the “−” sign of the area. Interchanging the columns of A we would move along the perimeter, with the parallelogram on our left: this would bring us to a positive area:
$$\begin{aligned} \det \left[ \begin{array}{c@{\quad }c} 2 &{} 1 \\ 4 &{} 3 \end{array} \right] =2 \end{aligned}$$ - 25.
One of the Authors (LP), when young and silly, tried to compute a determinant of order 7 by hand, with no success. But, at that time, no computer was available.
- 26.
“Line” means row or column, as you prefer.
- 27.
In a rigorous textbook, instead of “divided by...” you would read “multiplied by the reciprocal of...”, but this textbook is deliberately non-rigorous.
- 28.
See, for instance, the Example 1.2.5 on p. 10.
- 29.
This happens frequently when constructing simple models to illustrate pieces of some Social Science.
- 30.
It is the algorithm currently used by the most efficient software packages.
- 31.
The AA. would call it random number, but the prevailing nomenclature is adopted in this textbook.
- 32.
The order requirement is not necessary, but helpful in managing several applications.
- 33.
In principle, the possible values could be an infinity:
$$\begin{aligned} x_{1}, x_{2},\ldots , x_{n},\ldots \end{aligned}$$(1.6.1)To handle this case, we should introduce appropriate mathematical tools, like the sum of an infinity of addenda, called series. We have decided to omit this theme, thinking that models of the type (1.6.1) can be substituted for our readers with models:
$$\begin{aligned} x_{1}, x_{2},\ldots , x_{n} \end{aligned}$$with n large.
- 34.
The subtle reason for finding first a row vector and then a column vector, will turn out to be clear quickly.
- 35.
In reality the problem of transforming frequency into probability is very delicate, but it is not central for this textbook.
- 36.
Imagine you observe several times random quantities \(\mathbf {X}_{1},\mathbf {X }_{2},\ldots ,\mathbf {X}_{n}\) with probability distribution:
$$\begin{aligned} \left\{ \begin{array}{cl} \mathbf {x} &{} \leftarrow \text { possible values} \\ &{} \\ \mathbf {p}^{\text {T}} &{} \leftarrow \text { frequency} \end{array} \right. \end{aligned}$$Under broad conditions the (random) average of the observations:
$$\begin{aligned} \mathbf {Y}_{n}=\frac{\sum _{s=0}^{n}\mathbf {X}_{s}}{n} \end{aligned}$$approaches indefinitely \(\mathrm {E}\left( \mathbf {X}\right) \).
- 37.
Usually called ‘fat one’.
- 38.
Such a matrix (with all 0’s outside the principal diagonal) is called diagonal matrix. we think it is generated by the n-vector \(\mathbf {p }\), putting its components in the principal diagonal of a square matrix of order n. Frequent symbols employed for this transformation are:
$$\begin{aligned} P= \text {diag}\left( \mathbf {p}\right) =\widehat{\mathbf {p}} \end{aligned}$$ - 39.
It can take values only in the interval \( \left[ -1,1\right] \). When its value is \(\pm 1\), there is an exact linear affine relationship between the variables. When it takes values close to \( \pm 1\) such a relationship is only approximate. Its square \(\left( r\left[ \mathbf {X},\mathbf {Y}\right] \right) ^{2}\) is called determination coefficient and has an exciting interpretation. Assume that \(r=0.9\) and, therefore, \(r^{2}=0.81\). If the two statistical variables are income level and consumption level this means that the differences in income levels explain \(81\%\) of the variability (variance) of consumption levels.
- 40.
This model is due to Jan Tinbergen, 1969 Nobel Prize for Economics.
- 41.
In other words, a law that associates to each n-vector exactly one m-vector.
- 42.
It is easy to show that if there is more than one vector satisfying (1.8.1), then there is an infinity of vectors with that property. Let \(\mathbf {x}^{*}\) and \(\mathbf {x}^{**}\) be two different solutions:
$$\begin{aligned} A\mathbf {x}^{*}=\mathbf {b}\text { and }A\mathbf {x}^{**}=\mathbf {b} \text { } \end{aligned}$$take any two numbers \(\alpha \) and \(\left( 1-\alpha \right) \) adding up to 1. The vector \(\mathbf {v}\left( \alpha \right) =\alpha \mathbf {x}^{*}+\left( 1-\alpha \right) \mathbf {x}^{**}\) is another solution for every \(\alpha \):
$$\begin{aligned} A\mathbf {v}\left( \alpha \right)&=A\left[ \alpha \mathbf {x}^{*}+\left( 1-\alpha \right) \mathbf {x}^{**}\right] = \\&=\alpha A\mathbf {x}^{*}+\left( 1-\alpha \right) A\mathbf {x}^{**}= \\&=\alpha \mathbf {b+}\left( 1-\alpha \right) \mathbf {b=b} \end{aligned}$$ - 43.
Please, pay attention to the fact that a solution is a vector with n components. The solution is the vector. The components constitute the solution, but a single component is not a solution. Several times we met students that tried to convince us that each component of a vector \(\mathbf {x }^{*}\) is a solution. They did not succeed.
- 44.
The choice of the letter “\(\rho \)” has been made with hindsight.
- 45.
Gabriel Cramer (Genève, July 31, 1704 – Bagnols-sur-Cèze, January 4, 1752) - Professor of Philosophy and Mathematics at Geneva University. He got a PhD at 18.
- 46.
The null coefficients do come from the O blocks, while \(\beta \ne 0\), comes from \(\varvec{\beta }^{2}\ne \mathbf {o}\).
- 47.
The inverse:
$$\begin{aligned} \left[ \begin{array}{c@{\quad }c} 2 &{} 1 \\ 3 &{} 0 \end{array} \right] ^{-1} \end{aligned}$$could be obtained using determinants. Please note the recurrence of 3 in some denominators. It depends on the fact that the determinant of the (uncomplete) coefficient matrix is \(-3\). See above the Example 1.8.3 on p. 73.
- 48.
What happens if the dimension is greater than n is irrelevant.
- 49.
Trivially: the growth rate \(\lambda -1=\) \(10\%\) implies the growth factor \(\lambda =1+10\%=1.1\).
- 50.
It’s trivial, but frequently ignored: if \(\mathbf {x=o}\), the equality \(A \mathbf {x}=\lambda \mathbf {x}\) is obviously satisfied for every \(\lambda \). The economic interpretation of this mathematical fact is obvious: an economy without any capital stock can grow at any rate, which will bring it to... zero-stock everywhere.
- 51.
Please note that tr\(\left( A\right) =1+3=4\) and that \(\det A=3\).
- 52.
All the entries under the principal diagonal are null.
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Peccati, L., D’Amico, M., Cigola, M. (2018). Linear Algebra. In: Maths for Social Sciences. UNITEXT(), vol 113. Springer, Cham. https://doi.org/10.1007/978-3-030-02336-2_1
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DOI: https://doi.org/10.1007/978-3-030-02336-2_1
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