Website owner: James Miller
THE CONCEPTS OF LINEAR ALGEBRA REPRESENT THE END RESULT OF A SERIES OF GENERALIZATIONS AND ABSTRACTIONS
Linear Algebra contains certain fundamental concepts. Each of these fundamental concepts has gone through a series of generalizations and abstractions before reaching their final form in the very abstract subject of Linear Algebra. In Linear Algebra these fundamental concepts are all defined in very abstruse, abstract form as sets of postulates which, on first encounter, seem like intellectual nonsense. The definitions appear to make no sense, to be gobbledegook. They give no understanding of the concepts they represent. They don't explain or give any intuitive insight or feeling for the concepts. They are worded in such an abstract, unspecific, general way -- stating that that which is being defined is anything under the sun that has certain abstruse properties or that obeys certain very abstract, odd-sounding laws. Their wording sends one's mind into a dither. To understand the concepts you must have prior knowledge of the ideas from which they arose. You must understand the initial idea and then the series of generalizations and abstractions that led to the final concept. The final postulational forms of the definitions are really just lists of very abstract properties of the concepts, lists of abstract laws that they obey. Let us name these fundamental concepts of which we speak:
- concept of a vector
- concept of the length of a vector
- concept of the dot product, or inner product, of two vectors
- concept of a vector space
- concept of a subspace
- concept of a basis for a space or subspace
- concept of the dimension of a space or subspace
- concept of a linear transformation
To understand these concepts you must understand them in their initial form in two and three dimensional space. We encounter vectors initially in physics as quantities that have both magnitude and direction, quantities representing things like force, velocity, etc.. We view them as arrows in two or three dimensional space representing some vector quantity. Each arrow is defined by a doublet of numbers in two dimensional space, by a triplet of numbers in three dimensional space. In two or three dimensional space the concept of the length of a vector is a natural one and easily understood. Yet when this concept is encountered in Linear Algebra in the form of the concept of a "norm" it is unrecognizable. The definition in terms of postulates is mentally impenetrable. The same goes for the concept of the dot product and all the other concepts. They all go through a series of re-definitions, generalizations and abstractions, and in the end become unrecognizable. First the concept of a vector is re-defined as being simply an n-tuple. Then the concept of Euclidean n-dimensional space is defined in terms of an extension or generalization of the concept of three dimensional space. Then the concepts of vector length, dot product, etc. are defined for n-tuples. Each re-definition involves some alteration from the previous concept. In general, each re-definition involves the previous definition as a special case. Thus, in the evolution of the concepts, the previous forms of the concepts are included as special cases. But as the concepts become broader and more abstract, the previous versions of the concept become unrecognizable and the definitions become opaque and abstruse.
To really understand the concepts of a vector space, a subspace, a basis for a space or subspace, and the dimension of a space or subspace, for example, one must look to their meaning in two or three dimensional space. Once he has a good, intuitive understanding of them there he must be aware of various theorems which state various abstract properties they possess, various laws they obey. Then he is in a position to read the abstruse postulational definitions of Linear Algebra and make some sense of them remembering that they are merely lists of properties that the basic concepts in two, three and n dimensional space possess. For example: The final abstract definition encountered in Linear Algebra for a vector space is that anything under the sun that possesses a certain stated set of properties is a linear space. The same type definition is stated for a linear transformation. It is left up to your imagination to think of things that might possess those properties.
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