Math 304–504Linear AlgebraLecture 11:Vector spaces and their subspaces.Vector spaceVector 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 multiplicationA1. 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 denoted 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 + rb 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.Additional properties of vector spaces• The zero vector is unique.• For any a ∈ V , the negative −a is unique.• a + b = c ⇐⇒ a = c − b for all a, b, c ∈ V .• a + c = b + c ⇐⇒ a = b for all a, b, c ∈ V .• 0a = 0 for any a ∈ V .• (−1)a = −a for any a ∈ V .Examples of vector spacesIn most examples, addition and scalar multiplicationare natural operations so that properties A1–A8 areeasy to verify.• R: real numbers• Rn(n ≥ 1): coordinate vectors• C: complex numbers• Mm,n(R): m ×n matrices with real entries(also denoted Rm×n)• R∞: infinite sequ ences (x1, x2, . . . ), xi∈ R• {0}: the trivial vector spaceFunctional vector spaces• P: polynomials p(x) = a0+ a1x + · · · + anxn•ePn: polynomials of degree n (not a vector space)• Pn: polynomials of degree at most n• F (R): all functions f : R → R• C (R): all continuous functions f : R → R• F (R) \ C (R): all discontinuous func tionsf : R → R (not a vector space)• C1[a, b]: all continuously differentiable functionsf : [a, b] → R• C∞[a, b]: all smooth functions f : [a, b] → RCounterexample: 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.Counterexample: lazy scalingConsider the set V = Rnwith the standardaddition and a nonstandard scalar multiplication:r ⊙ a = a for any a ∈ Rnand r ∈ R.Properties A1–A4 hold because they do not involvescalar multiplication.A5. r ⊙ (a + b) = r ⊙ a + r ⊙ b ⇐⇒ a + b = a + bA6. (r + s) ⊙ a = r ⊙ a + s ⊙ a ⇐⇒ a = a + aA7. (rs) ⊙ a = r ⊙ (s ⊙ a) ⇐⇒ a = aA8. 1 ⊙ a = a ⇐⇒ a = aThe only property that fails is A6.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.• P: polynomials p(x) = a0+ a1x + · · · + anxn• Pn: polynomials of degree at most nPnis a subspace of P.• F (R): all functions f : R → R• C (R): all continuous functions f : R → RC (R) is a subspace of F (R).If S is a subset of a vector space V then S inheritsfrom V addition and scalar multiplication. HoweverS need not be closed under these operations.Proposition A subset S of a vector space V is asubspace of V if and only if S is nonempty andclosed under linear operations, i.e.,x, y ∈ S =⇒ x + y ∈ S,x ∈ S =⇒ rx ∈ S for all r ∈ R.Proof: “only if” is obvious.“if”: properties like associative, commutative, or distributivelaw hold for S because they hold for V . We only need toverify properties A3 and A4. Take any x ∈ S (note that S isnonempty). Then 0 = 0x ∈ S. Also, −x = (−1)x ∈ S.System of linear equations:a11x1+ a12x2+ · · · + a1nxn= b1a21x1+ a22x2+ · · · + a2nxn= b2· · · · · · · · ·am1x1+ am2x2+ · · · + amnxn= bmAny solution (x1, x2, . . . , xn) is an element of Rn.Theorem The solution set of the system is asubspace of Rnif and only if all equations in thesystem are homogeneous (all bi= 0).Theorem The solution set of the system is asubspace of Rnif and only if all equations in thesystem are homogeneous (all bi= 0).Proof: “only if”: the zero vector 0 = (0, 0, . . . , 0) is asolution only if all equations are homogeneous.“if”: if all equations are homogeneous then the solution set isnot empty because it contains 0.Suppose x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) aresolutions. That is, for every 1 ≤ i ≤ mai 1x1+ ai 2x2+ · · · + ainxn= 0,ai 1y1+ ai 2y2+ · · · + ainyn= 0.Then ai 1(x1+ y1) + ai 2(x2+ y2) + · · · + ain(xn+ yn) = 0and ai 1(rx1) + ai 2(rx2) + · · · + ain(rxn) = 0 for all r ∈ R.Hence x + y and r x are also solutions.Let V be a vector space and v1, v2, . . . , vn∈ V .Consider the set L of all linear combinationsr1v1+ r2v2+ · · · + rnvn, where r1, r2, . . . , rn∈ R.Theorem L is a subspace of V .Proof: First of all, …
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