Fundamentals of Linear Algebra

J.S. Chahal

• Publisher: Chapman and Hall/CRC
• Year: 2018
• ISBN:
  9780429758102
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques générales

Fundamentals of Linear Algebra is like no other book on the subject. By following a natural and unified approach to the subject it has, in less than 250 pages, achieved a more complete coverage of the subject than books with more than twice as many pages. For example, the textbooks in use in the United States prove the existence of a basis only for finite dimensional vector spaces. This book proves it for any given vector space.

With his experience in algebraic geometry and commutative algebra, the author defines the dimension of a vector space as its Krull dimension. By doing so, most of the facts about bases when the dimension is finite, are trivial consequences of this definition. To name one, the replacement theorem is no longer needed. It becomes obvious that any two bases of a finite dimensional vector space contain the same number of vectors. Moreover, this definition of the dimension works equally well when the geometric objects are nonlinear.

Features:

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• Presents theories and applications in an attempt to raise expectations and outcomes

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• The subject of linear algebra is presented over arbitrary fields

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• Includes many non-trivial examples which address real-world problems

1

Preliminaries

1.1 What Is Linear Algebra?

Manipulating matrices is not what linear algebra is all about. The matrices are only a convenient tool to represent and keep track of linear maps. And that too when the domain (and hence the range also) of such a function is finite dimensional. In its full generality, linear algebra is the study of functions in the most general context, which behave like the real valued function

$f\left(x\right)=mx$

(1.1)

of real variable x. The graph of (1.1) is a straight line through the origin with fixed slope m. Hence the name linear algebra.

The function y = f(x) = mx has a defining property: For constants c₁, c₂,

$f\left(c_1{x}_1+c_2{x}_2\right)=c_{1}f\left(x_1\right)+c_{2}f\left(x_2\right),$

(1.2)

which is equivalent to the following two conditions:

1) f(x₁ + x₂) = f(x₁) + f(x₂)

2) f(cx) = cf(x).

This is to say that any real valued function f(x) of a real variable with the property (1.2) has to be as in (1.1). In fact if f(1) = m, then by condition 2), f(x) = f(x1) = xf(1) = mx.

The notation y = f(x) for a function is not adequate, unless one says y = f(x) is a real valued function of a real variable x. A better and informative way to write it is f : ℝ → ℝ. In (1.1), the domain, which is the real line ℝ, is a 1-dimensional space. The values are also in the 1-dimensional space ℝ. If ℝ² is the plane consisting of points (x, y) and ℝ³ is the 3-dimensional space consisting of points P = (x, y, z), we can add and scale points in ℝⁿ (n = 2 or 3), considered as vectors. Thus we can also consider functions F : ℝ³ → ℝ² and call them linear if they have the property

$F\left(c_1{P}_1+c_2{P}_2\right)=c_{1}F\left(P_1\right)+c_{2}F\left(P_2\right)$

(1.3)

similar to property (1.2) of the function f(x) = mx. In general, one needs to study functions F : V → W, where V, W are spaces in the most general sense and the property (1.3) still makes sense. The spaces V, W that arise in various contexts will be defined in a unified way and will be called vector spaces or more appropriately, linear spaces. The functions F : V → W having the property (1.3) are linear maps, linear transformations, or simply linear. In this book, we study such spaces V, W and the linear maps F : V → W.

After making it precise what is meant by the dimension of a vector space, we shall show that if a vector space is finite dimensional, its elements are column vectors in a frame of reference to be called a basis. Moreover, if V and W are both finite dimensional and F : V → W is linear, then

$F\left(\mathit{x}\right)=M\mathit{x}$

(1.4)

where M is a matrix. The matrix M is obtained in a way similar to the 1 × 1 matrix M = (m) in the 1-dimen...