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Introduction to Abstract Algebra and Vector Spaces

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Introduction to Abstract Algebra and Vector Spaces Introduction to Abstract Algebra and Vector Spaces Vectors in Geometry A vector space is an algebraic structure that generalizes the geometrical concept of vector. The reader has hopefully been introduced to vectors in a previous math or physics course. We now review some of the basic properties of geometrical vectors that we desire to capture in the algebraic definition of a vector space. A geometric vector is an entity that has both a direction and magnitude (see Figure ). If we place a vector in the Cartesian plane with its tail at the origin, then the vector's direction and magnitude is completely determined by the location of its head in the plane, which can be represented with a point (x; y). This is considered one of the great achievements of the philosopher and mathematician Rene Descartes, who showed how a geometrical object, such as a vector, can be algebraically represented, in this case using a pair of numbers (x; y). Using the Cartesian representation of a vector, the following properties of vectors become more apparent. Addition and Subtraction Two vectors u = (x1; y1) and v = (x2; y2) can be added to form the new vector u+v = (x1 +x2; y1 +y2), and subtracted to form the new vector u−v = (x1 −x2; y1 −y2). Figure 1: A geometrical vector 1 Notice that vector addition is associative: u + (v + w) = (u + v) + w since the performed component-wise additions are all associative. It is also commutative: u + v = v + u. Zero Vector The vector 0 = (0; 0) represents the zero vector which has zero magnitude, and hence no direction. Additive Inverse Associated with each vector v = (x; y) is its additive inverse −v = (−x; −y), which has the same magnitude as v, but has the opposite direction. Moreover, v + (−v) = 0. Scaling Given real number r, every vector v = (x; y) can be scaled by r to form the new vector r · v = r(x; y) = (rx; ry). If r > 0, then r · v has the same direction as v, but its magnitude has been multiplied by a factor of r. If r = 0, then r · v = 0, where as r < 0 multiplies the magnitude of v by jrj, and r · v has the same direction as −v. Magnitude The magnitude of a vector v = (x; y) can be computed using the Pythagorean Theorem as jvj = px2 + y2. Angle Between Vectors If Θ(v) denotes the counterclockwise angle that v makes with the x-axis, then the angle between two vectors u and v is equal to jΘ(u) − Θ(v)j. Example 1. Demonstrate the above vector properties with vectors in the Cartesian plane. 2 Abelian Groups We now move from geometry to algebra. The simplest kind of algebraic structure is a set A, along with an operation that is closed under A, and that obeys a set of properties. For example, let A = I be the set of integers, and consider the addition operation +. This operation is closed under A since the addition of two integers results in another integer. Moreover, addition satisfies the following properties. Given arbitrary integers x, y, and z, then Associative x + (y + z) = (x + y) + z Commutative x + y = y + x Zero Element x + 0 = 0 + x = x Additive Inverse there exists an integer −x for which x + (−x) = 0 We can write this algebraic structure as a pair: (I; +), where the first component is the set, and the second is the operation that is defined and closed over the set. It turns out that (I; +) is not the only algebraic structure that satisfies the above four properties. For example (Q; +) and (R; +) also satisfy these properties, where Q is the set of rational numbers (i.e. fractions of the form p=q where p and q 6= 0 are integers), and R is the set of real numbers. In general, any algebraic structure that satisfies the above four properties is called an abelian group, while the above four properties referred to as the abelian-group axioms. Thus, when a mathematician says \such and such is an abelian group", she means that \such and such" is referring to a set, along with an addition operation closed over the set that is associative, commutative, allows for additive inverses, and for which the set has a zero element with respect to the addition. Once a class, such as abelian groups, of algebraic structures has been defined, it is the job the algebraist to study the logical implications of an algebraic structure that belongs in the class. For example, if an algebraic structure is an abelian group, then what other properties, if any, must it satisfy? Suppose an algebraist proves that all abelian groups have property P . Then she has not only proved that (I; +), (Q; +), and (R; +) have property P , but has also proved it for any other algebraic structure that meets the criteria for being an abelian group. 3 Example 2. Let R2 denote the set of ordered pairs (x; y), where x and y are real numbers. R2 is 2 referred to as the Cartesian plane. Verify that (R ; +) is an abelian group, where, for u = (x1; y1) and v = (x2; y2), u + v = (x1 + x2; y1 + y2). Example 3. Let (R−0; ·) be an algebraic structure, where R−0 is the set of real numbers, excluding 0, and · is the multiplication operation for real numbers. Verify that (R − 0; ·) is an abelian group. 4 Example 4. Let (F(R; R); +) be an algebraic structure, where F(R; R) is the set of functions f : R ! R having the real numbers as its domain and co-domain. and, given functions f; g 2 F(R; R), f +g is the function for which (f +g)(x) = f(x)+g(x), for all real numbers x. Prove that (F(R; R); +) is an abelian group. Example 5. Let (M2;2; ·) be an algebraic structure, where M2;2 is the set of 2 × 2 matrices having real-valued entries, and · represents matrix multiplication. Show that (M2;2; ·) is not an abelian group. However, show that (M2;2; +) is an abelian group, where + is matrix addition. 5 Vector Spaces An algebraic structure can have more than one operation. For example, the set of integers has both addition and multiplication operations which can be brought together to form the algebraic structure (I; +; ·). For this structure, the addition operation satisfies all the abelian-group axioms. Moreover, the multiplication operation is associative, commutative, and has 1 as its multiplicative identity. Finally, addition and multiplication, together, satisfy the distributive property x · (y + z) = x · y + x · z: It is these properties that depend on both operations that make the algebraic structure interesting to study. In this section we study a class of algebraic structures called vector space. Like integer arithmetic, a vector space consists of a set V and two operations that are closed over V. The first operation is addition, and it satisfies all the abelian-group axioms. In other words, if we ignore the second operation, then the algebraic structure (V; +) is an abelian group. Thus, every vector space is an abelian group. The second vector space operation is called the scaling operation, and is usually given the multi- plication symbol ·. This operation is described as follows. left input The left input is a real number r. right input The right input is a vector v 2 V. output The output is denoted r · v, or simply rv, and is a vector in V. The left input r has the effect of scaling the right input v. For example, if r = 2, then 2v is a vector that has the same \direction" as v, but has doubled in \magnitude" (here we place \direction" and \magnitude" in quotes, because we do not yet know what these terms mean in an abstract vector space). For this reason r is referred to as a scalar. With the terminology now in place, a vector space is an algebraic structure (V; +; ·), where (V; +) is an abelian group, and · is a scaling operation that satisfies the following properties. In what follows, r and s are arbitrary scalars, while u and v are arbitrary vectors. Moreover, e.g., r · v is expressed as rv. Unit Scalar 1v = v Associative r(sv) = (rs)v Distributive over V r(u + v) = ru + rv Distributive over R (r + s)v = rv + sv 6 These properties will henceforth be referred to as the scalar axioms. The scalar axioms together with the abelian-group axioms form what are called the vector-space axioms. Note that an element of a vector space is referred to as a vector. Example 6. We show that (R; +; ·), is a vector space, where R is the set of real numbers, and + and · are the usual addition and multiplication operations over R. Certainly, (R; +) is an abelian group since real-number addition is associative and commutative, 0 is the additive identity, and every real has an additive inverse. Moreover, · is a scaling operation, since its left input is a real, while its right input and output are both real (in this case the scalar set R is equal to the vector space set R). All that remains to show is that the scaling operation satisfies the scalar axioms. But notice that, since the scaling operation is just multiplication of two real numbers, these axioms are just the axioms of real-number multiplication, meaning that i) 1 is a multiplicative identy for R, and that ii) real-number multiplication is associative, commutative, and distributive. Example 7. Let (R2; +; ·) be an algebraic structure, where (R2; +) is the abelian group from Example 2, and · is the scaling operation defined as follows.
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