This is an example I made as a teacher. The idea was to take the vector and matrix away from the usual application in space to something more everyday: the kitchen. It is a humorous application of mathematical thinking to baking cakes. It touches the essentials of Linear algebra, without getting too formal.

It can also be found on my automatic exercise correcting, course building site called mamchecker based on google appengine. But here I have extended the idea a little bit.

Vectors, Tensors and Matrices in the Kitchen

Vectors

Let's talk about the ingredients of a cake, like eggs.

Eggs is a variable. We can take one egg, or two eggs, or three eggs,... The values are exclusive. We cannot take two eggs and three eggs for the same cake.

variable = set of exclusive values in a context

Three eggs. Three is the number. Egg gives meaning to the number. Egg is the unit. Eggs is a coordinate system (CS), that gives meaning to the numbers.

Three is the contravariant part. Egg, the unit, is the covariant part, of eggs. Note that we could take another unit, like sixpacks of eggs. Then the covariant part is bigger, but the contravariant part becomes smaller. This is already a coordinate transformation.

0.5 kg of flour. Another ingredient, another variable.

One variable is also called one-dimensional (1D) space. Two variables, from which we can choose independently, make up a two-dimensional space (2D), and so on.

Why introduce another word dimension and not just continue with variables? Actually, it's just to stick a little with conventional usage. Dimensions and variables are the same here.

A set of values from one or more variables is called a vector

• if these values can be added independently, i.e. eggs with eggs, ...
• and if they can be multiplied with numbers. I we add twice the same vector v we should be able to say 2v . Then all components of the vector are multiplied by 2.

The cake ingredients are a vector space over the real numbers ℝ .

Eggs, Flour, Butter, Sugar, ... they are not exclusive when baking a cake. Then they are not values of a variable, they are just a set. Let i be any of them and C be a cake. Cⁱ is the contravariant number value to the covariant unit C_i . i is the index that indicates, which ingredient.

C = CⁱC_i = ∑CⁱC_i

∑ : If there are two same indices, then these are summed by Einstein's convention.

This is like writing 'Ten Eggs and one unit of flour and ...`.

If we do that a lot, it becomes tiresome. We tend to make a table then. If we always make the same table, we remember the positions of eggs and flour, ... (position coding) and only write the numbers. A vector, though, is not just the numbers. It is the real thing, the cake recipe, in this example.

dot product

Eggs have nothing in common with flour, well, at least if we don't dig too deep, because they surely have carbon atoms in common. But for our kitchen thinking they have nothing in common. In mathematics this is orthogonal.

There is an operation between two vectors, that tells about how much they have in common: the inner product or dot product or skalar product. It's result is a scalar, i.e. a number. One denotes it by a dot in between, or by  < Egg|Flour >  .  < Egg|Egg >  = 1 , because they have all in common.  < Egg|Flour >  = 0 , because they have nothing in common (orthogonal).

In a cake recipe the units of ingredients are all orthogonal to each other: C_i⋅C_j = 0 , if j ≠ j . I used the dot notation here. They are also of unit 1, which makes them orthonormal.

The ingredients are the basis of the cake. The unit ingredients are the orthonormal basis vectors.

Though convenient it is not a necessity that the basis vectors have nothing in common, i.e. are orthogonal.

For an example let's move from the ingredients of cakes (ingredient vector space) to the cake vector space. Every cake is a unit to sell or bring to a party. And every cake is an ingredient vector, an independent choice from more variables. Cake A and cake B surely have ingredients in common. So these unit vectors in the cake vector space are not orthogonal to each other. The dot product is not 0.

AⁱA_i⋅B^jB_j = AⁱBⁱ ≠ 0

All the terms with i ≠ j are 0, because of the orthonormal basis. So they have been dropped after the first  =  . Then we have the usual formula for the dot product.

If A_i⋅B_j were not 0 for i ≠ j , we would have a A_i⋅B_j , called curvature tensor. As you can see it results from vectors, but cannot be added any more. That is the reason for the different name.

Coordinate Transformation

A vector in the cake vector space - How many of each kind of cake? Let's call it a cake assortment - can be transformed to the ingredient vector space by multiplying with a matrix. Every matrix column is the recipe of one kind of cake. The columns are the basis vectors of the cake space. By multiplying the assortment vector with the transformation matrix, we do a linear combination of the cake vectors.

A = A^jA_j = A^jC_jⁱC_i = CⁱC_i

The transformation matrix C_jⁱ says how much of the i ingredient cake j needs. j is the lower index and is a column. i is the upper index and it is a row. C_i is Egg, Flour, ... A^j is the amount of cake A_j in the assortment A . With A_j and C_i implicit, the transformation is:

Cⁱ = C_jⁱA^j

The summation over the j index is the matrix multiplication. It results in a number in each row telling about the total amount of one ingredient, like eggs.

On a matrix row there is an amount of an ingredient, like egg, for every kind of cake. This is the dot product  < egg|cake_j >  . In general, if we transform from a CS with basis vectors A_j to a CS with basis vectors C_i the transformation matrix is C_i⋅A_j , j being the columns.

A = A^jA_j = A^j < C_i|A_j > C_i

Inverse

If the number of i’s and the number of j’s are equal it is possible to find A^j from Cⁱ , the number of each kind of cake in the assortment, from the total amount of ingredients used.

In this example, though, the cake space and ingredient space do normally not have the same number of variables (number of variables = dimension). If we can have 10 ingredients and 3 kinds of cakes. Then the transformation matrix is 10x3 (10 rows, 3 columns). Such a m × n matrix with m ≠ n cannot be inverted, i.e. one cannot infer from the ingredients how many of each kind of cake are baked. Said differently: Not for every combination of ingredients there is a combination of cakes, which needs exactly this amount of ingredients.

Note

A non-square matrix can be pseudo-inverted, though: Moore-Penrose Pseudoinverse. For this example multiplying an ingredient vector with the pseudo-inverse would produce a cake vector, which minimizes unused quantities of ingredients (Method of Least Squares) or makes best use of the ingredients (Maximum Entropy Method).

If we change from one vector space to another with same dimensions, then we can get back to the starting one by multiplying with the inverse matrix (A^− 1 ). A^− 1 matrix multiplied by A gives the identity matrix I . Calculating the inverse means solving a system of linear equations.

A⋅x  = y     A^− 1A⋅x  = A^− 1y     Ix  = A^− 1y     x  = A^− 1y

In order for the inverse to exist, in addition to being square, the columns/rows of the matrix must be linearly independent. If not, then that is like effectively being in a smaller matrix (rank of matrix). For the cake example this means that every kind of cake must have a different combination of ingredients, which is some extra information that distinguishes it from the others and that can be used to code something.

Note

A square matrix can be inverted, if columns (rows) cannot be expressed as linear combination of the others, i.e. the rank of the matrix is equal to its dimension.

One can calculate the inverse of a square Matrix by:

• leaving out the ij cross and calculate the determinant = Minor M_ij
• change the sign, if i + j is odd
• transpose, i.e. mirror at the main diagonal (compare below: ij for A and ji for M )
• divide everything by the determinant

Short:

(A^− 1)_ij = (1)/(det(A))( − 1)^i + jM_ji

Inverse of a 2x2 Matrix:

M_ij is the diagonally opposite number. Because of the transposing the numbers left bottom and right top (secondary diagonal) stay where they are, but the sign changes. At the main diagonal the numbers get swapped, but since i + j is even the sign does not change.

• Main diagonal  →  swap, keep sign
• Secondary diagonal  →  no swap, change sign

There are algorithms, though, that are more efficient to calculate the inverse. And, of course, we use a computer program to do such calculations.