MATH 311Topics in Applied MathematicsLecture 7:Vector spaces.Subspaces.Linear operations on vectorsLet x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) ben-dimensional vectors, and r ∈ R be a scalar.Vector sum: x + y = (x1+ y1, x2+ y2, . . . , xn+ yn)Scalar multiple: rx = (rx1, rx2, . . . , rxn)Zero vector: 0 = (0, 0, . . . , 0)Negative of a vector: −y = (−y1, −y2, . . . , −yn)Vector difference:x − y = x + (−y) = (x1− y1, x2− y2, . . . , xn− yn)Properties of linear operationsx + y = y + x(x + y) + z = x + (y + z)x + 0 = 0 + x = xx + (−x) = (−x) + x = 0r(x + y) = r x + r y(r + s)x = rx + sx(rs)x = r(sx)1x = x0x = 0(−1)x = −xLinear operations on matricesLet A = (aij) and B = (bij) be m×n matrices,and r ∈ R be a scalar.Matrix sum: A + B = (aij+ bij)1≤i≤m, 1≤j≤nScalar multiple: rA = (raij)1≤i≤m, 1≤j≤nZero matrix O: all entries are zerosNegative of a matrix: −A = (−aij)1≤i≤m, 1≤j≤nMatrix difference: A − B = (aij− bij)1≤i≤m, 1≤j≤nAs far as the linear operations are concerned,the m×n matrices have the same properties asmn-dimensional vectors.Vector space: informal descriptionVector space = linear space = a set V of objects(called vectors) that can be added and scaled.That is, for any u, v ∈ V and r ∈ R expressionsu + v and rushould make sense.Certain restrictions apply. For instance,u + v = v + u,2u + 3u = 5u.That is, addition and scalar multiplication in Vshould be like those of n-dimensional vectors.Vector space: definitionVector space is a set V equipped with twooperations α : V × V → V and µ : R × V → Vthat have certain properties (listed below).The operation α is called addition. For anyu, v ∈ V , the element α(u, v) is denoted u + v.The operation µ is called scalar multiplication. Forany r ∈ R and u ∈ V , the element µ(r, u) isdenoted ru.Properties of addition and scalar multiplication(brief)A1. a + b = b + aA2. (a + b) + c = a + (b + c)A3. a + 0 = 0 + a = aA4. a + (−a) = (−a) + a = 0A5. r(a + b) = r a + r bA6. (r + s)a = r a + saA7. (rs)a = r(sa)A8. 1a = aProperties of addition and scalar multiplication (detailed)A1. a + b = b + a for all a, b ∈ V .A2. (a + b) + c = a + (b + c) for all a, b, c ∈ V .A3. There exists an element of V , called the zerovector and de noted 0, such that a + 0 = 0 + a = afor all a ∈ V .A4. For any a ∈ V there exists an element of V ,denoted −a, such that a + (−a) = (−a) + a = 0.A5. r(a + b) = r a + r b for all r ∈ R and a, b ∈ V .A6. (r + s)a = r a + sa for all r , s ∈ R and a ∈ V .A7. (rs)a = r(sa) for all r, s ∈ R and a ∈ V .A8. 1a = a for all a ∈ V .• Associativity of addition implies that a multiplesum u1+ u2+ ··· + ukis well defined for anyu1, u2, . . . , uk∈ V .• Subtraction in V is defined as usual:a − b = a + (−b).• Addition and scalar multiplication are calledlinear operations.Given u1, u2, . . . , uk∈ V and r1, r2, . . . , rk∈ R,r1u1+ r2u2+ ··· + rkukis called a linear combination of u1, u2, . . . , uk.Examples of vector spacesIn most examples, addition and scalar multiplicationare natural operations so that properties A1–A8 areeasy to verify.• Rn: n-dimensional co ordinate vectors• Mm,n(R): m×n matrices with real entries• R∞: infinite sequences (x1, x2, . . . ), xi∈ RFor any x = (x1, x2, . . . ), y = (y1, y2, . . . ) ∈ R∞and r ∈ Rlet x + y = (x1+ y1, x2+ y2, . . . ), rx = (rx1, rx2, . . . ).Then 0 = (0, 0, . . . ) and −x = (−x1, −x2, . . . ).• {0}: the trivial vector space0 + 0 = 0, r0 = 0, −0 = 0.Functional vector spaces• F (R): the set of all functions f : R → RGiven functions f , g ∈ F (R) and a scalar r ∈ R, let(f + g )(x) = f (x) + g(x) and (rf )(x) = rf (x) for all x ∈ R.Zero vector: o(x) = 0. Negative: (−f )(x) = −f (x).• C (R): all continuous functions f : R → RLinear operations are inherited from F (R). We only need tocheck that f , g ∈ C (R) =⇒ f +g , rf ∈ C(R), the zerofunction is continuous, and f ∈ C (R) =⇒ −f ∈ C (R).• C1(R): all continuously differentiable functionsf : R → R• C∞(R): all smooth functions f : R → R• P: all polynomials p(x) = a0+ a1x + ··· + anxnSome general observations• The zero vector is unique.If z1and z2are zeros then z1= z1+ z2= z2.• For any a ∈ V , the negative −a is unique.Suppose b and b′are negatives of a. Thenb′= b′+ 0 = b′+ (a + b) = (b′+ a) + b = 0 + b = b.• 0a = 0 for any a ∈ V .Indeed, 0a + a = 0a + 1a = (0 + 1)a = 1a = a.Then 0a + a = a =⇒ 0a + a − a = a − a =⇒ 0a = 0.• (−1)a = −a for any a ∈ V .Indeed, a + (−1)a = (−1)a + a = (−1)a + 1a = (−1 + 1)a= 0a = 0.Counterexample: dumb scalingConsider the set V = Rnwith the standardaddition and a nonstandard scalar multiplication:r ⊙ a = 0 for any a ∈ Rnand r ∈ R.Properties A1–A4 hold because they do not involvescalar multiplication.A5. r ⊙ (a + b) = r ⊙ a + r ⊙ b ⇐⇒ 0 = 0 + 0A6. (r + s) ⊙ a = r ⊙ a + s ⊙ a ⇐⇒ 0 = 0 + 0A7. (rs) ⊙ a = r ⊙ (s ⊙ a) ⇐⇒ 0 = 0A8. 1 ⊙ a = a ⇐⇒ 0 = aA8 is the only property that fails. As a consequence,property A8 does not follow from properties A1–A7.Subspaces of vector spacesDefinition. A vector space V0is a subspace of avector space V if V0⊂ V and the linear operationson V0agree with the linear operations on V .Examples.• F (R): all functions f : R → R• C (R): all continuous functions f : R → RC (R) is a subspace of F (R).• P: polynomials p(x) = a0+ a1x …