Vector Spaces Every Student Should Know The following vector spaces are important examples every student should know after having taken a Þrst course in linear algebra. In particular, you should be able to verify that all eight properties of a vector space are satisÞed! In each case, we use and to denote addition and scalar multiplication merely to emphasize that these⊕ operations¯ do not necessarily correspond to the usual vector addition and scalar multiplication in Rn! Positive Real Numbers • Let R+ denote the positive real numbers. For x, y R+,deÞne x y = xy and ∈ ⊕ for c R,deÞne c x = xc. That is, to add two vectors (positive real numbers in this∈ case), we multiply¯ them as ordinary real numbers, and to perform scalar multiplication, we use exponentiation. 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a positive real number x,whatis x? − 4. What is a Þnite basis for this vector space? 5. What are some interesting subspaces? Rn • n For x, y R ,letx y = z where zi = xi + yi (componentwise addition), and ∈ ⊕ for c R, c x = z where zi = cxi. This vector space is the prototype vector space ∈from which¯ much of the theory has been developed. 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a vector x,whatis x? − 4. What is a Þnite basis for this vector space? (the unit vectors e1,e2,...,en , for example, is one) { } 5. What are some interesting subspaces? Cn • n For x, y C ,letx y = z where zi = xi + yi (componentwise addition), and ∈ ⊕ for c C, c x = z where zi = cxi. (Recall how addition and multiplication ∈ ¯ of complex number are deÞned: if a + bi, c + di C,then(a + bi)+(c + di)= (a + c)+(b + d)i and (a + bi)(c + di)=(ac bd)+(∈ ad + bc)i.) − 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a vector x,whatis x? − 4. What is a Þnite basis for this vector space? 5. What are some interesting subspaces? m n Matrices with Entries in R • × Let m,n(R) denote the set of all m n matrices with entries in R.For M × A, B m,n(R),deÞne A B = C where cij = aij +bij (componentwise addition) ∈ M ⊕ and for c R, c A = D where dij = caij. ∈ ¯ 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a matrix A,whatis A? − 4. What is a Þnite basis for this vector space? 5. What are some interesting subspaces? 2 Polynomials • Let ℘(R) denote the set of all polynomials in the variable t (or x, etc). For p(t),q(t) ℘(R),deÞne (p q)(t)=p(t)+q(t),andforc R,deÞne (c p)(x)= cp(x). That∈ is, to add two⊕ vectors (polynomials in this∈ case), we add¯ them as ordinary functions. To perform scalar multiplication, just multiply the polynomial through by the scalar. For example, (6t2 t +1) (t3 t2 +2t 1) = t3 +5t2 + t and 4 (t2 7t +3)=4t2 28t +12. − ⊕ − − ¯ − − 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a polynomial p(t),whatis( p)(t)? − 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? Polynomials with Degree n • ≤ Let ℘n(R) denote the set of all polynomials in the variable t having degree at most n.DeÞne and as for the vector space ℘(R). ⊕ ¯ 1. Are all the vector space properties satisÞed? 2. What is the 0 of this vector space? 3. For a polynomial p(t),whatis( p)(t)? − 4. What is a Þnite basis for this vector space? 5. What are some interesting subspaces? Continuous Functions on an Interval [a, b] or ( , + ) • −∞ ∞ Let [a, b] and ( , + ) denote the set of all continuous functions deÞned on a ÞxedC interval C[a,−∞ b] and∞( , + ) respectively. DeÞne and as for the −∞ ∞ ⊕ ¯ vector space ℘(R). 3 1. Are all the vector space properties satisÞed? In particular, are the closure properties satisÞed? 2. What is the 0 of these vector spaces? 3. For a continuous function f(x),whatis( f)(x)? − 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? (Note that ℘(R) and ℘n(R) are subspaces of ( , + ).) C −∞ ∞ Functions with Continuous First Derivatives on ( , + ) • −∞ ∞ Let 1( , + ) denote the set of all functions with continuous Þrst deriva- C −∞ ∞ tives on ( , + ).DeÞne and as for the vector space ℘(R). −∞ ∞ ⊕ ¯ 1. Are all the vector space properties satisÞed? In particular, are the closure properties satisÞed? 2. What is the 0 of this vector space? 3. For a function f(x) 1( , + ),whatis( f)(x)? ∈ C −∞ ∞ − 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? (Note that ℘(R) and ℘n(R) and are subspaces of 1( , + ).) C −∞ ∞ Smooth Functions on ( , + ) • −∞ ∞ Let ∞( , + ) denote the set of all functions with continuous derivatives of all ordersC −∞ on ( ∞, + ). This class of functions are often referred to as smooth −∞ ∞ functions. DeÞne and as for the vector space ℘(R). ⊕ ¯ 1. Are all the vector space properties satisÞed? In particular, are the closure properties satisÞed? 2. What is the 0 of this vector space? 3. For a function f(x) ∞( , + ),whatis( f)(x)? ∈ C −∞ ∞ − 4 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? (Note that ℘(R) and ℘n(R) are subspaces of ∞( , + ).) C −∞ ∞ Linear Transformations From Rn to Rm • Let (Rn, Rm) denote the set of all linear transformations from Rn to Rm. L n m (This vector space is important in advanced algebra.) For T1,T2 (R , R ), ∈ L deÞne (T1 T2)(x)=T1(x)+T2(x) ,andforc R,deÞne (c T1)(x)=cT1(x) ⊕ n ∈ ¯ m (note that x is a vector in R and T1(x) and T2(x) are vectors in R ). 1. Are all the vector space properties satisÞed? In particular, are the closure properties satisÞed? 2. What is the 0 of this vector space? 3.ForalineartransformationT (Rn, Rm),whatis( T )(x)? ∈ L − 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? Linear Transformations From (V,+, ) to (W, , ) • · ¢ ¡ (This vector space is nothing more than a direct generalization of the vector n m space (R , R )). Let (V,+, ) and (W, ¢, ¡) be any two arbitrary vector spaces. Let (LV,W) denote the set of all· linear transformations from V to W.(Asbefore, L this vector space is important in advanced algebra.) For T1,T2 (V,W),deÞne ∈ L (T1 T2)(v)=T1(v) T2(v),andforc R,deÞne (c T1)(v)=c T1(v). ⊕ ¢ ∈ ¯ ¡ 1. Are all the vector space properties satisÞed? In particular, are the closure properties satisÞed? 2. What is the 0 of this vector space? 3.ForalineartransformationT (V,W),whatis( T )(x)? ∈ L − 4. What is a Þnite basis for this vector space? Does it have one? 5. What are some interesting subspaces? 5.