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4.1 Vectors in Rn Chapter 4: Vector Spaces. 4.1 Vectors in Rn. 4.1 Vectors in Rn. Objective: Represent a vector in the plane as a directed line segment. Objective: Perform basic vector operations in R2 and represent them graphically. Objective: Perform basic vector operations in Rn. Prove basic properties about vectors and their operations in Rn. In physics and engineering, a vector is an object with magnitude and direction and represented graphically by a directed line segment. In mathematics we have a much more general definition of a vector. Geometrically, a vector in the plane is represented by a directed line segment with its initial point at the origin and its terminal (final) point at (x1, x2). The same ordered pair used to represent the terminal point is used to represent the vector. That is, x = (x1, x2). The coordinates x1 and x2 are called the components of the vector x. Two vectors u = (u1, u2) and v = (v1, v2) are equal iff u1 = v1 and u2 = v2. Vector operations in R2 def Vector Addition: u + v = (u1, u2) + (v1, v2) (u1 + v1, u2 + v2). def Scalar Multiplication: cu = c(u1, u2) (cu1, cu2). Negative: –u (–u1, –u2). Notice that –u = (–1)u. Subtraction: u – v u + (–v) = (u1, u2) + (–v1, –v2) = (u1 – v1, u2 – v2). The zero vector in R2 is 0 = (0, 0). p. 73 Chapter 4: Vector Spaces. 4.1 Vectors in Rn. Examples Let u = (2, 4) and v = (–4, 1). Illustrate the following graphically. u = (2, 4), v = (–4, 1), u + v = (–2, 5), 1.5u = (3, 6), – v = (4, –1), v – u = (–6, –3) p. 74 Chapter 4: Vector Spaces. 4.1 Vectors in Rn. Theorem 4.1 Properties of Vector Addition and Scalar Multiplication in the Plane (R2) Let u, v, and w be vectors in R2, and let c and d be scalars. 1) u + v is a vector in R2. Closure under addition 2) u + v = v + u Commutative property of addition 3) (u + v) + w = u + (v + w) Associative property of addition 4) u + 0 = u Existence of additive identity 5) u + (–u) = 0 Existence of additive inverses 6) cv is a vector in R2. Closure under scalar multiplication 7) c(u + v) = cu + cv Distributive property over vector addition 8) (c + d)u = cu + du Distributive property over scalar addition 9) c(du) = (cd)u Associative property 10) 1u = u Multiplicative identity property Proof of (3): Associative property of addition (u + v) + w = [(u1, u2) + (v1, v2)] + (w1, w2) = (u1 + v1, u2 + v2) + (w1, w2) Definition of vector addition = ((u1 + v1) + w1, (u2 + v2) + w2) Definition of vector addition = (u1 + (v1 + w1), u2 + (v2 + w2)) Assoc. prop. of addition of real numbers = (u1, u2) + (v1 + w1, v2 + w2) Definition of vector addition = (u1, u2) + [(v1, v2) + (w1, w2)] Definition of vector addition = u + (v + w) p. 75 Chapter 4: Vector Spaces. 4.1 Vectors in Rn. Proof of (8): Distributive property of scalar multiplication over real number addition (c + d)u = (c + d)(u1, u2) = ((c + d)u1, (c + d)u2) Definition of scalar multiplication = (cu1 + du1, cu2 + du2) Distributive property of real numbers = (cu1, cu2) + (du1, du2) Definition of vector addition = c(u1, u2) + d(u1, u2) Definition of scalar multiplication = cu + du To add (1, 4) + (2, –2) in Mathematica, type {1,4}+{2,-2} You can also assign a variable by typing u={1,4} You can perform scalar multiplication by 3u or 3*u To add (1, 4) + (2, –2) on the TI-89, you can type 1,4+2,-2 or 14+2-2 You can also assign a variable by typing 1,4.U You can perform scalar multiplication by 3u or 3u Vector operations in Rn We can generalize from the 2-dimensional plane R2 to an n-space Rn of ordered n-tuples. For example, R1 = R = set of all real numbers; R2 = 2-space = set of all ordered pairs of real numbers; R3 = 3-space = set of all ordered triplesdef of real numbers.\ n An n-tuple (x1, x2, …, xn) can be viewed as a point in R with the xi as its coordinates, or as a vector with the xi as its components. The standard vector operations in Rn are Vector Addition: u + v = (u1, u2, …, un) + (v1, v2, …, vn) (u1 + v1, u2 + v2, …, un + vn). Scalar Multiplication: cu = c(u1, u2, …, un) (cu1, cu2, …, cun). Negative: –u (–u1, –u2, …, –un). Notice that –u = (–1)u. Subtraction: u – v u + (–v) = (u1, u2, …, un) + (–v1, –v2, …, –vn) = (u1 – v1, u2 – v2, …, un – vn). The zero vector in Rn is 0 = (0, 0, …, 0). p. 76 Chapter 4: Vector Spaces. 4.1 Vectors in Rn. Theorem 4.2 Properties of Vector Addition and Scalar Multiplication in the Plane (Rn) Let u, v, and w be vectors in Rn, and let c and d be scalars. 1) u + v is a vector in Rn. Closure under addition 2) u + v = v + u Commutative property of addition 3) (u + v) + w = u + (v + w) Associative property of addition 4) u + 0 = u Existence of additive identity 5) u + (–u) = 0 Existence of additive inverses 6) cv is a vector in Rn. Closure under scalar multiplication 7) c(u + v) = cu + cv Distributive property over vector addition 8) (c + d)u = cu + du Distributive property over scalar addition 9) c(du) = (cd)u Associative property 10) 1u = u Multiplicative identity property The vector 0 is called the additive identity in Rn and –v is the additive inverse of v. Theorem 4.3 Properties of Vector Addition and Scalar Multiplication in Rn Let v be a vector in Rn and let c be a scalar. Then 1) The additive identity is unique. That is, if v + u = v, then u = 0. 2) The additive inverse of v is unique. That is, if v + u = 0, then u = –v. 3) 0v = 0 4) c0 = 0 5) If cv = 0, then c = 0 or v = 0. 6) –(–v) = v Proof of (1): Uniqueness of the additive identity v + u = v Given (v + u) + (–v) = v + (–v) Add –v to both sides (u + v) + (–v) = v + (–v) Commutative property u + (v + (–v)) = v + (–v) Associative property u + 0 = 0 Additive inverse u = 0 Additive identity p. 77 Chapter 4: Vector Spaces. 4.1 Vectors in Rn. Proof of (2): Uniqueness of the additive inverse v + u = 0 Given (–v) + (v + u) = (–v) + 0 Add –v to both sides [(–v) + v] + u = (–v) + 0 Associative property 0 + u = (–v) + 0 Additive inverse u + 0 = (–v) + 0 Commutative property u = –v Additive identity p. 78 Chapter 4: Vector Spaces. 4.2 Vector Spaces. 4.2 Vector Spaces. Objective: Define a vector space and recognize some important examples of vector spaces. Objective: Show that a given set is not a vector space. (Optional) Theorem 4.2 listed ten properties of vector addition and scalar multiplication in Rn. However, there are many other sets (Cn, sets of matrices, polynomials, functions) besides Rn that can be given suitable definitions of vector addition and scalar multiplication so that they too satisfy the same ten properties. Hence, one branch of mathematics, linear algebra, can study all of these. Definition of a Vector Space Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the axioms listed below are satisfied for every u, v, and w in V and every scalar c and d in a given field F (usually, F = R or F = C), then V is called a vector space over F. 1) u + v is in V. Closure under addition 2) u + v = v + u Commutative property * 3) (u + v) + w = u + (v + w) Associative property 4) V has a zero vector 0 such that Existence of additive identity for every u in V, u + 0 = u 5) For every u in V, there is a vector Existence of additive inverses (opposites) denoted by –u such that u + (–u) = 0 6) cv is a vector in V. Closure under scalar multiplication 7) c(u + v) = cu + cv Distributive property over vector addition 8) (c + d)u = cu + du Distributive property over scalar addition 9) c(du) = (cd)u Associative property 10) 1u = u Scalar identity Notice that a vector space actually consists of four entities: a set V of vectors, a field F of scalars, and two defined operations (vector additiondef and scalar multiplication). Be sure all four entities are clearly understood. (For example, I could keep the set V of vectors, the field F of scalars, the same definition of scalar multiplication, but change the definition of how to add vectors and end up with a different vector space, or end up with something that is no longer a vector space.) Examples of Vector Spaces. (Unless otherwise stated, assume the field is R.) R2 with the standard operations * u + v = (u1, u2) + (v1, v2) (u1 + v1, u2 + v2). cu = c(u1, u2) (cu1, cu2). 0 = (0, 0) –u = (–u1, –u2) Rn with the standard operations. Note that this includes R2, which is just R with the usual addition and multiplication. Cn over the field C with the standard operations p. 79 Chapter 4: Vector Spaces. 4.2 Vector Spaces. More Examples of Vector Spaces. (Unless otherwise stated, assume the field is R.) The vector space M2,3of all 23 real matrices with the standard operations a11 a12 a13 b11 b12 b13 a11 b11 a12 b12 a13 b13 A + B = + = a21 a22 a23 b21 b22 b23 a21 b21 a22 b22 a23 b23 ca11 ca12 ca13 and cA = c = ca 21 ca 22 ca 23 The vector space Mm,n of all mn real matrices with the standard operations.
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