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Linear algebra is the study of systems of equations. In mathematics, equations are used to find the value of an unknown variable. For instance:

70 z + 07 = 7007

Solve for the unknown z

A system of equations is a set of equations that contains multiple variables. When one deals with such a system, one tries to find the values of the variables. For instance:

313 A1 + 956 A2 = 48

663 A1 – 1113 A2 = 12

Solve for the unknowns A1 and A2

In order to understand the spiritual reality of the science of linear algebra it is important to interpret mathematical equations in a spiritual way. What are we really trying to solve for when we are given an equation such 5x + 3 = 0? Obviously we want to solve for the unknown variable, but what does this unknown number represent?

Mathematical equations are the scientific manifestation of the most important spiritual question:

Who is God?

Answering this question is the divine purpose of creation and the techniques of solving mathematical equations are the manifestations of the spiritual techniques used in Sufism to reach the station of knowledge of God (marifat’ullah).

The divine purpose of creation is to know God through His Attributes, as He says in the Holy Quran and in traditional sayings:

and

Therefore, linear algebra, the study of linear systems of equations, is the study of the knowledge of God. In fact, the science of mathematics in general is a direct manifestation of the saying I was a Hidden Treasure and I wanted to be known.

From this saying we may understand two important points:

- God is unknown

- God created ways for creation to know about Him

The first point is manifested in mathematics as the unknown variable in a given equation. The second point is manifested in science as the set of fundamental laws of nature. In this chapter, this set is the science of linear algebra and the study of matrices.

Since the divine purpose of creation is eternal by nature, the question “who is God” is perpetually being asked by creation. This perpetual repetition is reflected in science as the infinite number of equations we as humans try to solve at every moment. We will always be able to set up new equations involving unknown variables because the answer to the question “who is God” will never be completely found as God possess an infinite number of Attributes.

It may appear as if we are “free” to invent new equations at any moment. However, our intellectual capacity to do so is not a sign of freedom. Rather, it is a sign of constraint because we are eternally bound to reflect the infinite number of appearances of the divine question through our intellectual abilities. This is a manifestation of the 3rd postulate known as the eternal constraint.

God may be known through His Attributes only and not through His Essence which is called Hu in Arabic. Therefore, it is impossible to find an answer to the question “who is Hu?”. We may, however, find relative answers to the question “who is God?” because the answers would be related to His Attributes.

Consequently, one may see that there are two faces to the divine question “who is God?”: one face that may be answered through God’s Attributes and one that may not be answered. These questions are related to the fundamental questions of linear algebra.

There are two fundamental questions in spirituality:

“Can we find an answer to the question “who is God?”

“If yes, is God manifesting Himself through His Encompassing Attributes or through their Uniqueness?

The first question means: “is the door of knowledge of God open?” The answer to such a question is positive if we are referring to the divine Attributes and is negative if we are referring to the divine Essence. In other words:

- Yes, we may know God through His Attributes

- No, we may not know God’s holy Essence

When the answer is positive, the unknown knowledge is revealed. Mathematically speaking, we say that the system of equations has a solution. When the answer is negative, the unknown knowledge is not disclosed and the system of equations has no solution.

The second question can only be answered if the answer to the first one is positive. In this case one attempts to get knowledge of the nature of the divine Attributes which could either be inclusive (i.e., encompassing) or exclusive (i.e., unique) as we saw in the previous chapter.

The two fundamental spiritual questions discussed above are manifested in linear algebra as the two fundamental questions of linear algebra. These questions are asked whenever one attempts to solve a system of questions. They are:

“Is the given system of equations consistent? (i.e., does the system have a solution?)

“If yes, is the solution unique, or is there an infinite number of solutions?”

The two fundamental questions gave birth to the two most important theorems of linear algebra: the existence theorem and the uniqueness theorem. We shall analyze these theorems in our upcoming discussion but let us first analyze the fundamental mathematical questions in the realm of spirituality.

Scientists call the first fundamental question (“does the system have a solution?) the question of existence. Mathematically speaking, a system is said to be consistent when the unknown variables are identifiable, that is, if it is possible to find a value for the variable x. Spiritually speaking this question represents our fundamental question of Knowledge: can we find an answer to the question “who is God?”.

If the answer is positive then one may gain access to some knowledge of the divine Attributes and the system of equations is said to have a solution. At this time, one may proceed to determine the type of solutions the system has: either an infinite number of solutions or a unique solution. This step represents the second fundamental question of linear algebra.

The second fundamental question (“Is the solution unique or is there an infinite number of solutions?”) deals with the nature of the divine Attributes. Mathematically speaking, the solution is unique if and only if there is only one possible value for the variable x. If the solution is not unique then the variable x may assume an infinite number of values.

Spiritually speaking, the uniqueness of the solution is a manifestation of the uniqueness of God’s Attributes while the multiplicity of the solutions represent the encompassing nature of His Attributes.

In the Holy Name of God, the All-Compassionate, the All-Merciful,

With peace and blessings upon His Prophet Muhammad SAW

upon his holy family, and upon his blessed Companions.

Bridging the Gap between Spirituality and Science