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 triples 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
def Vector Addition: u + v = (u1, u2, …, un) + (v1, v2, …, vn) (u1 + v1, u2 + v2, …, un + vn). def 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 addition 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 * def def 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.
The vector space P2 of all polynomials of degree 2 or less with the usual operations. 2 2 Let p(x) = a0 + a1x + a2x and q(x) = b0 + b1x + b2x . def Define the usual operations (p + q)(x) p(x) + q(x) and (cp)(x) c[p(x)]
We can verify closure under addition: 2 2 (p + q)(x) = p(x) + q(x) = a0 + a1x + a2x + b0 + b1x + b2x 2 = (a0 + b0) + (a1 + b1)x + (a2 + b2)x which is a polynomial of degree 2 or less (less if a2 + b2 = 0). Notice that we have used the commutative and distributive properties of real numbers. The other axioms can be verified in a similar manner. . Note that 0(x) = 0 + 0x + 0x2.
The vector space Pn of all polynomials of degree n or less with the usual operations.
The vector space P of all polynomials with the usual operations.
The vector space C(–,) of continuous real- valued functions on the domain (–,) For 8 example, x2 + 1, , sin(x), and ex are vectors 1 x 2 in this space. Addition and scalar multiplication are defined in the usual way. (f + g)(x) f(x) + g(x) and (cf )(x) c[f (x)]
f, g, and f + g are vectors in C(–,), just as u, v, and u + v and are vectors in Rn. f (x) can be
thought of as a component of f, just as ui is a component of u. u has n components: u1, u2, …, 2 un. f has an infinite number of components: …, f (–2), …, f ( 3 ), …, f (0), …, f ( ), ….
The additive identity (zero function) is f0(x) = 0 (the x-axis), and given f (x), the additive inverse of f is [–f ](x) = –[ f (x)].
p. 80 Chapter 4: Vector Spaces. 4.2 Vector Spaces. Another Example of a Vector Spaces . The vector space C[a, b] of continuous real-valued functions on the domain [a, b] over the field R.
The most important reason for defining an abstract vector space using the ten axioms above is that we can make general statements about all vector spaces. I.e. the same proof can be used for Rn and for C[a, b].
Theorem 4.4 Properties of Vector Addition and Scalar Multiplication
Let v be a vector in Vn and let c be a scalar. Then
1) 0v = 0 2) c0 = 0 3) If cv = 0, then c = 0 or v = 0. 4) –1v = –v
Proof of (2): c0 = 0
c0 = c(0 + 0) Additive identity
c0 = c0 + c0 Distributive property
c0 + –( c0) = (c0 + c0) + –( c0) Add + –( c0) to both sides
c0 + –( c0) = c0 + (c0 + –( c0)) Associative property
0 = c0 Additive inverse
Proof of (3): If cv = 0, then c = 0 or c 0. If c 0, then
cv = c0 Given
Multiply both sides by Multiplicative c–1cv = c–10 inverse in R
p. 81 Chapter 4: Vector Spaces. 4.2 Vector Spaces.
c(c–1)v = c–10 Commutative property of multiplication
1v = c–10 Multiplicative inverse in R
v c–10 Scalar identity
v = 0 Theorem 4.3(2) – just proved
Thus, either c = 0 or v = 0.
Examples that are not Vector Spaces
2 2
Z (ordered pair of integers) over the field R. Z is not closed under scalar multiplication, for 1 1 2 example 2 (1, 2) = ( 2 , 1) Z . Aside on notation: 1 Z means “1 is an element of (is a member) of the set of integers. optional Z means “1 is not an element of the set of integers. Although Z2 satisfied Axioms 1–5 and 10 of a vector space, it is not a vector space because not all axioms are satisfied. .() V = The set of second-degree polynomials is not a vector space because it is not closed under addition. For example, let p(x) = x2 and q(x) = –x2 + x + 1. Then p(x) + q(x) = x + 1 is a first degree polynomial.
Let V = R2 with the standard vector addition but nonstandard scalar multiplication defined by
c(u1, u2) = (cu1, 0). Show that V is not a vector space.
It turns out that the only axiom that is not satisfied in this case is (10) Scalar identity. For example, 1(2, 3) = (2, 0) (2, 3).
p. 82 Chapter 4: Vector Spaces. 4.2 Vector Spaces.
Another Example that is not a Vector Spaces
Rotations in three dimensions represented as arrows using the right-hand rule. The direction of the arrow represents the direction of the rotation, via the right-hand rule, while the length of the arrow represents the optional magnitude of the direction in degrees. Scalar multiplication is the standard operation (stretching the arrow, or reversing the direction if
the scalar is negative). “Vector addition” (e.g. ) is the first rotation followed by
the second. This is not a vector space because “vector addition” is not commutative.
.
(In Chapter 6, we will see that rotations can be represented not as vectors, but as matrices.)
p. 83