3 Systems of differential equations Theorem 3. Any n linearly independent vectors in an n dimensional space V must also span V. That is to say, any n linearly independent vectors in an n dimensional space V are a basis for V. PROOF. Let x 1, x2 , ... , xn be n linearly independent vectors in an n dimensional space V. To show that they span V, we must show that every x in V can be written as a linear combination of x 1,x2 , ... ,xn. Tothis end, pick any x in V and consider the set of vectors x,x 1,x2, ... ,xn. This is a set of ( n + 1) vectors in the n dimensional space V; by Lemma 2, they must be linearly dependent. Consequently, there exist constants c,c 1,c2, ... ,cn, not all zero, such that (8) Now ci'O, for otherwise the set of vectors x 1,x2, ... ,xn would be linearly dependent. Therefore, we can divide both sides of (8) by c to obtain that cl I c2 2 x=--x - - x c c Hence, any n linearly independent vectors in an n dimensional space V must also span V. D Example 7. Prove that the vectors form a basis for R2 Solution. To determine whether x 1 and x2 are linearly dependent or linearly independent, we consider the equation (9) Equation (9) implies that c 1 + c 2 = 0 and c 1 - c 2 = 0. Adding these two equations gives c 1 = 0 while subtracting these two equations gives c2 = 0. Consequently, x 1 and x2 are two linearly independent vectors in the two dimensional space R2 Hence, by Theorem 3, they must also span V. EXERCISES In each of Exercises 1-4, determine whether the given set of vectors is linearly dependent or linearly independent. 2. ( 288 D (n and ( ~) 3.3 4. Dimension of a vector space (i ), (-10, (-~) and (=D 5. Let V be the set of all 2 X 2 matrices. Determine whether the following sets of matrices are linearly dependent or linearly independent in V. (a) ( ~ (b)n g), (6 g), (6 n, (~ 6) n, (~ 6) and (g and (- ~ D -2) 1 . 6. Let V be the space of all polynomials in t of degree < 2. (a) Show that dimV=3. (b) Let Pt, p 2 and p 3 be the three polynomials whose values at any time t are (t-1f, (t-2) 2 , and (t-1) (t-2) respectively. Show thatpt>pz, andp 3 are linearly independent. Hence, conclude from Theorem 3 that Pt, p 2, and p 3 form a basis for V. 7. Let V be the set of all solutions of the differential equation d 2y / dt 2 - y (a) Show that V is a vector space. (b) Find a basis for V. =0. 8. Let V be the set of all solutions of the differential equation (dyjdt 3)+y=O which satisfy y(O) = 0. Show that V is a vector space and find a basis for it. 9. Let V be the set of all polynomialsp(t)=a0 +att+a2 t 2 which satisfy p(O) +2p'(O) + 3p"(O) =0. Show that V is a vector space and find a basis for it. 10. Let V be the set of all solutions of the differential equation x=(~ Show that -116 )x. 1 6 form a basis for V. 289 3 Systems of differential equations 11. Let V be a vector space. We say that W is a subspace of Vif W is a subset of V which is itself a vector space. Let W be the subset of R3 which consists of all vectors x= (~~) which satisfy the equations x 1 +x2 +2x3 =0 2x 1 - x 2 + x 3 =0 6x 1 +6x3 =0. Show that W is a subspace of R3 and find a basis for it. 12. Prove that any n vectors which span an n dimensional vector space V must be linearly independent. Hint: Show that any set of linearly dependent vectors contains a linearly independent subset which also spans V. 13. Let v1, v2, , v" be n vectors in a vector space V. Let W be the subset of V which consists of alllinear combinations c 1v1 +c2r+ ... +cnv" of v1,r, ... ,v". Show that W is a subspace of V, and that dim W..; n. 14. Let V be the set of all functions f(t) which are analytic for JtJ m. Show that we can find vectors vm+ 1, , vn so that v1, v2, , v"', yn+ 1, , v" form a basis for V. That is to say, any set of m linearly independent vectors in an n > m dimensional space V can be completed to form a basis for V. 16. Find a basis for R3 which includes the vectors (l) 17. (a) Show that are linearly independent in R3 . (b) Let 290 and ( 3.4 Applications of linear algebra to differential equations Since v1,v2, and v3 are linearly independent they are a basis and x=y 1v1 + y 2 v2 + y 3 v3 What is the relationship between the original coordinates X; and the new coordinates y1? (c) Express the relations between coordinates in the form x=By. Show that the columns of 8 are v1, v2 , and v3 3.4 Applications of linear algebra to differential equations Recall that an important tool in solving the second-order linear homogeneous equation (d 2y I dt 2) + p(t)(dy I dt) + q(t)y = 0 was the existenceuniqueness theorem stated in Section 2.1. In a similar manner, we will make extensive use of Theorem 4 below in solving the homogeneous linear system of differential equations dx =Ax dt , A= ~~II {1) an I The proof of this theorem will be indicated in Section 4.6. Theorem 4 (Existence-uniqueness theorem). There exists one, and only one, solution of the initial-value problern x? dx =Ax dt , x{t 0)=x0 = Moreover, this solution exists for - oo X~ {2) < t < oo. Theorem 4 is an extremely powerful theorem, and has many implications. In particular, if x(t) is a nontrivial solution, then x(t)*O for any t. (If x(t*)=O for some t*, then x(t) must be identically zero, since it, and the trivial solution, satisfy the same differential equation and have the same value at t = t* .) Wehave already shown (see Example 7, Section 3.2) that the space V of all solutions of (1) is a vector space. Our next step is to determine the dimension of V. Theorem 5. The dimension of the space V of all solutions of the homogeneaus linear system of differential equations (I) is n. PROOF. We will exhibit a basis for V which contains n elements. To this 291 3 Systems of differential equations Example 7. Let V be the set of all vector-valued solutions x(t)= of the vector differential equation (2) x=Ax, V is a vector space under the usual Operations of vector addition and scalar multiplication. To wit, observe that axioms (i), (ii), and (v)-(viii) are automatically satisfied. Hence, we need only verify that (a) The sum of any two solutions of (2) is again a solution. (b) A constant times a solution of (2) is again a solution. (c) The vector-valued function x(t)= x 1 ~t) =[~] xn(t) 0 is a solution of (2) (axiom (iii)). (d) The negative of any solution of (2) is again a solution (axiom (iv)). Now (a) and (b) are exactly Theorem I of the previous section, while (d) is a special case of (b). To verify (c) we observe that Hence the vector-valued function x(t)=:O is always a solution of the differential equation (2). ExERCISEs In each of Problems 1-6, determine whether the given set of elements form a vector space under the properties of vector addition and scalar multiplication defined in Section 3.1. 278 3.3 Dimension of a vector space 1. The set of all elements x= ( ~~) where 3x -2x =0 2. The set of all elements x = ( :~ ) where x + x + x = 0 3. The set of all elements x = ( :~ ) where xf + xi + xj = 1 4. The set of all elements x = ( ~~) where x 2 1 1 3 2 1 + x2 + x3 = 1 5. The set of elements x = ( %) for all real numbers a and b 6. The set of all elements x = ( x 1 +x2 +x3 =0, :i) where x 1 -x2 +2x3 =0, 3x 1 -x2 +5x3 =0 In each of Problems 7-11 determine whether the given set of functions form a vector space under the usual operations of function addition and multiplication of a function by a constant. 7. The set of all polynomials of degree .;;; 4 8. The set of all differentiable functions 9. The set of all differentiable functions whose derivative at t = 1 is three 10. The set of all solutions of the differential equationy"+y=cost 11. The set of all functionsy(t) which have period 2'll', that isy(t+2'll')=y(t) 12. Show that the set of all vector-valued solutions x(t)=(x 1 (t)) x 2 (t) of the system of differential equations dx 1 - =x2 + 1 ' dt is not a vector space. 3.3 Dimension of a vector space Let V be the set of all solutions y(t) of the second-order linear homogeneous equation (d 2y I dt 2 ) + p(t)(dy I dt) + q(t)y =0. Recall that every solutiony(t) can be expressedas a linear combination of any two linearly independent solutions. Thus, if we knew two "independent" functions y 1( t) and 279