# EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)

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1 EXERCISE SET The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily verified. Axiom 4 holds with = (, ) and Axiom 5 holds with (x, y) = ( x 2, y 2). Axiom 7 fails because k((x, y) + (x, y )) = k(x + x +, y + y + ) = (kx + kx + k, ky + ky + k ) while k(x, y) + k(x, y ) = (kx, ky) + (kx, ky ) = (kx + kx +, ky + ky + ) Hence, k(u + v) = ku + kv only if k =. Axiom 8 also fails, since if u = (x, y), then (k + )u = ((k + )x, (k + )y) but ku + u = (kx, ky) + ( x, y) = ((k + )x +, (k + )y + ). This is a vector space. Axioms 2, 3, 7, 8, 9, and follow from properties of matrix addition and scalar multiplication. We verify the remaining axioms. () If we add two matrices of this form, the result will again be a matrix of this form: (*) a c b + a c d = + b+ d 29

2 22 Exercise Set 5. (4) The 2 2 zero matrix is of the appropriate form and has the desired properties. (5) If u is a matrix of the given form, then = a u b is again of the desired form and u + ( u) = ( u) + u =. (6) If u is any matrix of this form, then ku is (**) k a ka b = kb and ku has the desired form.. This is a vector space. We shall check only four of the axioms because the others follow easily from various properties of the real numbers. () If f and g are real-valued functions defined everywhere, then so is f + g. We must also check that if f() = g() =, then (f + g)() =. But (f + g)() = f() + g() = + =. (4) The zero vector is the function z which is zero everywhere on the real line. In particular, z() =. (5) If f is a function in the set, then f is also in the set since it is defined for all real numbers and f() = =. Moreover, f + ( f) = ( f) + f = z. (6) If f is in the set and k is any real number, then kf is a real valued function defined everywhere. Moreover, kf() = k =. 2. This is a vector space and the proof is almost a direct repeat of that for Problem. In fact, we need only modify the two equations (*) and (**) in the following way: a a+ b c c d a c a c a+ b b + + c+ d d = + ( + )+ b+ d ( a+ c)+ ( b+ d) b+ d a a+ b k a+ b b = ka ka + kb ka + kb kb Note that if u is any matrix of this form, then ( ) u = a ( a+ b) ( a b) b +

3 Exercise Set This is a vector space with = (, ) and x = (, x). The details are easily checked. 5. We must check all ten properties: () If x and y are positive reals, so is x + y = xy. (2) x + y = xy = yx = y + x (3) x + (y + z) = x(yz) = (xy)z = (x + y) + z (4) There is an object, the positive real number, which is such that + x = x = x = x = x + for all positive real numbers x. (5) For each positive real x, the positive real /x acts as the negative: x + (/x) = x(/x) = = = = (/x)x = (/x) + x (6) If k is a real and x is a positive real, then kx = x k is again a positive real. (7) k(x + y) = (xy) k = x k y k = kx + ky (8) (k + )x = x k+ = x k x = kx + x (9) k( x) = ( x) k = ( x) k = x k = x k = (k )x () x = x = x 6. This is not a vector space, since properties (7) and (8) fail. It is easy to show, for instance, that ku + kv = k 2 (u + v). Property (4) holds with = (, ), but Property (5) fails because, for instance, (, ) has no inverse. 7. (a) Only Axiom 8 fails to hold in this case. Let k and m be scalars. Then (k + m)(x, y, z) = ((k + m) 2 x, (k + m) 2 y, (k + m) 2 z) = (k 2 x, k 2 y, k 2 z) + (2kmx, 2kmy, 2kmz) + (m 2 x, m 2 y, m 2 z) = k(x, y, z) + m(x, y, z) + (2kmx, 2kmy, 2kmz) k(x, y, z) + m(x, y, z), and Axiom 8 fails to hold. (b) Only Axioms 3 & 4 fail for this set. Axiom 3: Using the obvious notation, we have u + (v + w) = (u, u 2, u 3 ) + (v 3 + w 3, v 2 + w 2, v + w ) = (u 3 + v + w, u 2 + v 2 + w 2, u + v 3 + w 3 ) whereas (u + v) + w = (u 3 + v 3, u 2 + v 2, u + v ) + (w, w 2, w 3 ) = (u + v + w 3, u 2 + v 2 + w 2, u 3 + v 3 + w )

4 222 Exercise Set 5. Thus, u + (v + w) (u + w) + w. Axiom 4: There is no zero vector in this set. If we assume that there is, and let = (z, z 2, z 3 ), then for any vector (a, b, c), we have (a, b, c) + (z, z 2, z 3 ) = (c + z 3, b + z 2, a + z ) = (a, b, c). Solving for the z i s, we have z 3 = a c, z 2 = and z = c a. Thus, there is no one zero vector that will work for every vector (a, b, c) in R 3. (c) Let V be the set of all 2 2 invertible matrices and let A be a matrix in V. Since we are using standard matrix addition and scalar multiplication, the majority of axioms hold. However, the following axioms fail for this set V: Axiom : Clearly if A is invertible, then so is A. However, the matrix A + ( A) = is not invertible, and thus A + ( A) is not in V, meaning V is not closed under addition. Axiom 4: We ve shown that the zero matrix is not in V, so this axiom fails. Axiom 6: For any 2 2 invertible matrix A, det(ka) = k 2 det(a), so for k, the matrix ka is also invertible. However, if k =, then ka is not invertible, so this axiom fails. Thus, V is not a vector space. a 8. Let V be the set of all matrices of the form. We will verify each of the vector b a b c space axioms using matrices A =, B =, and C =, and scalars a2 b c 2 2 k and m. () We can see from the given formula that the sum of two matrices in V is again a matrix in V. Thus, Axiom holds. a+ b b+ a (2) A + B = = = B + A. a2 + b2 b2 + a2 (3) We have a A+ B+ C = ( ) + a2 b+ c b + c 2 2 = a + b + c a + b + c a+ b = a + b c c 2 = ( A+ B) + C and Axiom 3 is satisfied.

5 Exercise Set (4) The zero vector in this space is the matrix =. Then a A + = a 2 a + = A = + a2 + And similarly, + A = A. Thus, Axiom 4 holds. a a (5) If A =, then A =, and we have a a 2 2 a a A + ( A) = + = = a2 a2 Similarly, we have ( A) + A =. (6) We can see from the given formula that if A is in V, then ka is also a matrix in V. Thus, Axiom 6 holds. (7) We have a+ b k( A+ B) = a + b 2 2 = ka ( + b) ka ( 2 + b) 2 ka + kb = ka + kb 2 2 = ka ka 2 kb + kb 2 = ka + kb and Axiom 7 is satisfied. (8) We have ( ) k+ m a ( k+ m) A= ( ) k+ m a ka ma = + ka2 + ma2 2 ka ma = + ka2 ma 2 = ka + ma and Axiom 8 holds.

6 224 Exercise Set 5. (9) We have kma ( ( ))= k ma = ma2 ( ) k ma ( ) k ma 2 km a ( ) = ( ) = ( km) A km a2 so Axiom 9 holds. () Finally, we have a A = a 2 a = a 2 a = a2 = A and Axiom is also satisfied. 9. (a) Let V be the set of all ordered pairs (x, y) that satisfy the equation ax + by = c, for fixed constants a, b and c. Since we are using the standard operations of addition and scalar multiplication, Axioms 2, 3, 5, 7, 8, 9, will hold automatically. However, for Axiom 4 to hold, we need the zero vector (, ) to be in V. Thus a + b = c, which forces c =. In this case, Axioms and 6 are also satisfied. Thus, the set of all points in R 2 lying on a line is a vector space exactly in the case when the line passes through the origin. (b) Let V be the set of all ordered triples (x, y, z) that satisfy the equation ax + by + cz = d, for fixed constants a, b, c and d. Since we are using the standard operations of addition and scalar multiplication, Axioms 2, 3, 5, 7, 8, 9, will hold automatically. However, for Axiom 4 to hold, we need the zero vector (,, ) to be in V. Thus a + b + c = d, which forces d =. In this case, Axioms and 6 are also satisfied. Thus, the set of all points in R 3 lying on a plane is a vector space exactly in the case when the plane passes through the origin. 2. Let V be the set of all 2 2 invertible matrices. With the given operations, V is not a vector space. We will check all axioms: () The product of two invertible matrices is invertible, so Axiom holds. (2) In general, matrix multiplication is not commutative, so AB BA and Axiom 2 fails. (3) Matrix multiplication is associative so A(BC) = (AB)C and Axiom 3 holds. (4) The identity matrix functions as the zero vector here: AI = IA = A and Axiom 4 holds. (5) The inverse of A functions as A in this case: A(A ) = A A = I, so Axiom 5 holds.

7 Exercise Set (6) If k =, then ka is not invertible, so ka is not in V and Axiom 6 fails. (7) Axiom 7 fails: k(ab) (ka)(kb). (8) Axiom 8 fails as well: (k + m)a (ka)(ma). (9) Since we are using regular scalar multiplication, Axiom 9 holds. () Again, since we are using regular scalar multiplication, Axiom holds. Thus, V is not a vector space since Axioms 2, 6, 7, & 8 fail. (Only one failure is necessary.) 22. Properties (2), (3), and (7) () all hold because a line passing through the origin in 3- space is a collection of triples of real numbers and the set of all such triples with the usual operations is a vector space. To verify the remaining four properties, we need only check that () If u and v lie on the line, so does u + v. (4) The vector (,,) lies on the line (which it does by hypothesis, since the line passes through the origin). (5) If u lies on the line, so does u. (6) If u lies on the line, so does any real multiple ku of u. We check (), leaving (5) and (6) to you. The line passes through the origin and therefore has the parametric equations x = at, y = bt, z = ct where a, b, and c are fixed real numbers and t is the parameter. Thus, if u and v lie on the line, we have u = (at, bt, ct ) and v = (at 2, bt 2, ct 2 ). Therefore u + v = (a(t + t 2 ), b(t + t 2 ), c(t + t 2 )), which is also on the line. 25. No. Planes which do not pass through the origin do not contain the zero vector. 26. No. The set of polynomials of exactly degree does not contain the zero polynomial, and hence has no zero vector. 27. Since this space has only one element, it would have to be the zero vector. In fact, this is just the zero vector space. 28. If a vector space V had just two distinct vectors, and u, then we would have to define vector addition and scalar multiplication on V. Theorem 5.. ensures that ku unless k =. Thus we would have to define ku = u for all k. But then we would have u = 2 u = ( + )u = u + u = u + u,

8 226 Exercise Set 5. so that u + u = u. However, u = ( )u = u + ( )u = u + ( )u = which implies that ( )u u. Hence ( )u =, which is contrary to Theorem 5... Thus V cannot be a vector space. 3. We are given that ku =. Suppose that k. Then k (ku) = k k u = ()u = u (Axioms 9 and ) But k (ku) = k = (By hypothesis and Part (b)) Thus u =. That is, either k = or u =. 32. Suppose that there are two zero vectors, and 2. If we apply Axiom 4 to both of these zero vectors, we have = + 2 = 2 Hence, the two zero vectors are identical. 33. Suppose that u has two negatives, ( u) and ( u) 2. Then ( u) = ( u) + = ( u) + (u + ( u) 2 ) = (( u) + u) + ( u) 2 = + ( u) 2 = ( u) 2 Axiom 5 guarantees that u must have at least one negative. We have proved that it has at most one. 34. Following the hint, we have (u + v) (v + u) = (u + v) + ( (v + u)) by Theorem 5.. = (u + v) + (( )v + ( )u) by Property (7) = (u + v) + (( v) + ( u)) by Theorem 5.. = ((u + v) + ( v)) + ( u) by Property (3) = (u + (v + ( v)) + ( u) by Property (3) = (u + ) + ( u) by Property (5) = u + ( u) by Property (4) = by Property (5)

9 Exercise Set Thus (u + v) + ( )(v + u) =, from which it follows that so that (u + v) + ( (v + u)) = by Theorem 5.. [(u + v) + ( (v + u))] + (v + u) = + (v + u) adding to both sides of the equation and thus (u + v) + ( (v + u) + (v + u)) = + (v + u) by Property (3) or (u + v) + = + (v + u) by Property (5) so that finally u + v = v + u by Property (4).

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11 EXERCISE SET 5.2. (a) The set is closed under vector addition because (a,, ) + (b,, ) = (a + b,, ) It is closed under scalar multiplication because k(a,, ) = (ka,, ) Therefore it is a subspace of R 3. (b) This set is not closed under either vector addition or scalar multiplication. For example, (a,, ) + (b,, ) = (a + b, 2, 2) and (a + b, 2, 2) does not belong to the set. Thus it is not a subspace. (c) This set is closed under vector addition because (a, b, ) + (a 2, b 2, ) = (a + a 2, b + b 2, ). It is also closed under scalar multiplication because k(a, b, ) = (ka, kb, ). Therefore, it is a subspace of R (a) This set is closed under vector addition since the sum of two integers is again an integer. However, it is not closed under scalar multiplication since the product ku where k is real and a is an integer need not be an integer. Thus, the set is not a subspace. (c) If det(a) = det(b) =, it does not necessarily follow that det(a + B) =. For instance, let A = and B =. Thus the set is not a subspace. 229

12 23 Exercise Set 5.2 (e) This set is closed under vector addition because a a a b b a b a b a + b b = + + a b a b a b a b = + + ( a+ b) a+ b ( ) It is also closed under scalar multiplication because k a a a a = ka ka = k( a) k( a) ka ka. ( ka) ( ka) Therefore, it is a subspace of M (a) This is the set of all polynominals with degree 3 and with a constant term which is equal to zero. Certainly, the sum of any two such polynomials is a polynomial with degree 3 and with a constant term which is equal to zero. The same is true of a constant multiple of such a polynomial. Hence, this set is a subspace of P 3. (c) The sum of two polynomials, each with degree 3 and each with integral coefficients, is again a polynomial with degree 3 and with integral coefficients. Hence, the subset is closed under vector addition. However, a constant multiple of such a polynomial will not necessarily have integral coefficients since the constant need not be an integer. Thus, the subset is not closed under scalar multiplication and is therefore not a subspace. 4. (a) The function f(x) = for all x belongs to the set, but the function ( )f(x) = for all x does not. Hence, the set is not closed under scalar multiplication and is therefore not a subspace. (c) Suppose that f and g are in the set. Then (f + g)() = f() + g() = = 4 and 2f() = ( 2)(2) = 4 This set is therefore not closed under either operation. (e) Let f(x) = a + b sin x and g(x) = c + d sin x be two functions in this set. Then (f + g)(x) = (a + c) + (b + d) sin x

13 Exercise Set and k(f(x)) = ka + kb sin x Thus, both closure properties are satisfied and the set is a subspace. 5. (b) If A and B are in the set, then a ij = a ji and b ij = b ji for all i and j. Thus a ij + b ij = (a ji + b ji ) so that A + B is also in the set. Also a ij = a ji implies that ka ij = (ka ji ), so that ka is in the set for all real k. Thus the set is a subspace. (c) For A and B to be in the set it is necessary and sufficient for both to be invertible, but the sum of 2 invertible matrices need not be invertible. (For instance, let B = A.) Thus A + B need not be in the set, so the set is not a subspace. 6. (b) The matrix reduces to 2 so the solution space is the line x = 2t, y = t, z =. (d) The matrix reduces to the identity matrix, so the solution space is the origin. (f) The matrix reduces to 3 so the solution space is the plane x 3y + z =. 7. (a) We look for constants a and b such that au + bv = (2, 2, 2), or a(, 2, 2) + b(, 3, ) = (2, 2, 2) Equating corresponding vector components gives the following system of equations: b = 2 2a+ 3b = 2 2a b = 2

14 232 Exercise Set 5.2 From the first equation, we see that b = 2. Substituting this value into the remaining equations yields a = 2. Thus (2, 2, 2) is a linear combination of u and v. (c) We look for constants a and b such that au + bv = (, 4, 5), or a(, 2, 2) + b(, 3, ) = (, 4, 5) Equating corresponding components gives the following system of equations: b = 2a + 3b = 4 2a b = 5 From the first equation, we see that b =. If we substitute this value into the remaining equations, we find that a = 2 and a = 5/2. Thus, the system of equations is inconsistent and therefore (, 4, 5) is not a linear combination of u and v. 8. (a) We look for constants a, b, and c such that au + bv + cw = ( 9, 7, 5); that is, such that a(2,, 4) + b(,, 3) + c(3, 2, 5) = ( 9, 7, 5) If we equate corresponding components, we obtain the system 2a + b + 3c = 9 a b + 2c = 7 4a + 3b + 5c = 5 The augmented matrix for this system is The reduced row-echelon form of this matrix is 2 2 Thus a = 2, b =, and c = 2 and ( 9, 7, 5) is therefore a linear combination of u, v, and w.

15 Exercise Set (c) This time we look for constants a, b, and c such that au + bv + cw = (,, ) If we choose a = b = c =, then it is obvious that au + bv + cw = (,, ). We now proceed to show that a = b = c = is the only choice. To this end, we equate components to obtain a system of equations whose augmented matrix is From Part (a), we know that this matrix can be reduced to Thus, a = b = c = is the only solution. 9. (a) We look for constants a, b, and c such that ap + bp 2 + cp 3 = 9 7x 5x 2 If we substitute the expressions for p, p 2, and p 3 into the above equation and equate corresponding coefficients, we find that we have exactly the same system of equations that we had in Problem 8(a), above. Thus, we know that a = 2, b =, and c = 2 and thus 2p + p 2 2p 3 = 9 7x 5x 2. (c) Just as Problem 9(a) was Problem 8(a) in disguise, Problem 9(c) is Problem 8(c) in different dress. The constants are the same, so that = p + p 2 + p 3.. (a) We ask if there are constants a, b, and c such that 4 2 a b c = 6 8 8

16 234 Exercise Set 5.2 If we multiply, add, and equate corresponding matrix entries, we obtain the following system of equations: 4a + b = 6 b + 2c = 8 2a + 2b + c = 2a + 3b + 4c = 8 This system has the solution a =, b = 2, c = 3; thus, the matrix is a linear combination of the three given matrices. (b) Clearly the zero matrix is a linear combination of any set of matrices since we can always choose the scalars to be zero.. (a) Given any vector (x, y, z) in R 3, we must determine whether or not there are constants a, b, and c such that (x, y, z) = av + bv 2 + cv 3 = a(2, 2, 2) + b(,, 3) + c(,, ) = (2a, 2a + c, 2a + 3b + c) or x = 2a y = 2a + c z = 2a + 3b + c This is a system of equations for a, b, and c. Since the determinant of the system is nonzero, the system of equations must have a solution for any values of x, y, and z, whatsoever. Therefore, v, v 2, and v 3 do indeed span R 3. Note that we can also show that the system of equations has a solution by solving for a, b, and c explicitly. (c) We follow the same procedure that we used in Part (a). This time we obtain the system of equations 3a + 2b + 5c + d = x a 3b 2c + 4d = y 4a + 5b + 9c d = z

17 Exercise Set The augmented matrix of this system is x y z which reduces to y x 3y z 4y x 3y 7 Thus the system is inconsistent unless the last entry in the last row of the above matrix is zero. Since this is not the case for all values of x, y, and z, the given vectors do not span R (a) Since cos(2x) = () cos 2 x + ( ) sin 2 x for all x, it follows that cos(2x) lies in the space spanned by cos 2 x and sin 2 x. (b) Suppose that 3 + x 2 is in the space spanned by cos 2 x and sin 2 x; that is, 3 + x 2 = a cos 2 x + b sin 2 x for some constants a and b. This equation must hold for all x. If we set x =, we find that a = 3. However, if we set x = π, we find a = 3 + π 2. Thus we have a contradiction, so 3 + x 2 is not in the space spanned by cos 2 x and sin 2 x. 3. Given an arbitrary polynomial a + a x + a 2 x 2 in P 2, we ask whether there are numbers a, b, c and d such that a + a x + a 2 x 2 = ap + bp 2 + cp 3 + dp 4 If we equate coefficients, we obtain the system of equations: a = a + 3b + 5c 2d a = a + b c 2d a 2 =2a + 4c + 2d

18 236 Exercise Set 5.2 A row-echelon form of the augmented matrix of this system is a a + a 4 a + 3a+ 2 a2 Thus the system is inconsistent whenever a + 3a + 2a 2 (for example, when a =, a =, and a 2 = ). Hence the given polynomials do not span P (a) As before, we look for constants a, b, and c such that (2, 3, 7, 3) = av + bv 2 + cv 3 If we equate components, we obtain the following system of equations: 2a + 3b c = 2 a b = 3 5b + 2c = 7 3a + 2b + c = 3 The augmented matrix of this system is This reduces to 2 Thus a = 2, b =, and c =, and the existence of a solution guarantees that the given vector is in span {v, v 2, v 3 }.

19 Exercise Set (c) Proceeding as in Part (a), we obtain the matrix This reduces to a matrix whose last row is [ ]. Thus the system is inconsistent and hence the given vector is not in span {v, v 2, v 3 }. 5. The plane has the vector u v = (, 7, 7) as a normal and passes through the point (,,). Thus its equation is y z =. Alternatively, we look for conditions on a vector (x, y, z) which will insure that it lies in span {u, v}. That is, we look for numbers a and b such that (x, y, z) = au + bv = a(,, ) + b(3, 4, 4) If we expand and equate components, we obtain a system whose augmented matrix is x y z This reduces to the matrix 3 x x+ y 7 y+ z 7 y+ z Thus the system is consistent if and only if = or y = z The set of solution vectors of such a system does not contain the zero vector. Hence it cannot be a subspace of R n.

20 238 Exercise Set Suppose that span{v, v 2,, v r } = span{w, w 2,, w k }. Since each vector v i in S belongs to span{v, v 2,, v r }, it must, by the definition of span{w, w 2,, w k }, be a linear combination of the vectors in S. The converse must also hold. Now suppose that each vector in S is a linear combination of those in S and conversely. Then we can express each vector v i as a linear combination of the vectors w, w 2,, w k, so span{v, v 2,, v r } span{w, w 2,, w k }. But conversely we have span{w, w 2,, w k } span{v, v 2,, v r }, so the two sets are equal. 9. Note that if we solve the system v = aw + bw 2, we find that v = w + w 2. Similarly, v 2 = 2w + w 2, v 3 = w + w 2, w = v + v 2 v 3, and w 2 = v + v 2 + v (a) We simply note that the sum of two continuous functions is a continuous function and that a constant times a continuous function is a continuous function. (b) We recall that the sum of two differentiable functions is a differentiable function and that a constant times a differentiable function is a differentiable function. 23. (a) False. The system has the form Ax = b where b has at least one nonzero entry. Suppose that x and x 2 are two solutions of this system; that is, Ax = b and Ax 2 = b. Then A(x + x 2 ) = Ax + Ax 2 = b + b b Thus the solution set is not closed under vector addition and so cannot form a subspace of R n. Alternatively, we could show that it is not closed under scalar multiplication. (b) True. Let u and v be vectors in W. Then we are given that ku + v is in W for all scalars k. If k =, this shows that W is closed under addition. If k = and u = v, then the zero vector of V must be in W. Thus, we can let v = to show that W is closed under scalar multiplication. (d) True. Let W and W 2 be subspaces of V. Then if u and v are in W W 2, we know that u + v must be in both W and W 2, as must ku for every scalar k. This follows from the closure of both W and W 2 under vector addition and scalar multiplication. (e) False. Span{v} = span{2v}, but v 2 v in general. 24. (a) Two vectors in R 3 will span a plane if and only if one is not a constant multiple of the other. They will span a line if and only if one is a constant multiple of the other. (b) Span{u} = span{v} if and only if u is a nonzero multiple of v. Why? (c) The solution set will be a subspace of R n if and only if b =. See Exercise 23(a).

21 Exercise Set No. For instance, (, ) is in W and (, ) is in W 2, but (, ) + (, ) = (2, ) is in neither W nor W (a) The matrices and span M 22.,, 27. They cannot all lie in the same plane.

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23 EXERCISE SET (a) Clearly neither of these vectors is a multiple of the other. Thus they are linearly independent. (b) Following the technique used in Example 4, we consider the system of linear equations 3k + 5k 2 + k 3 = k 2 + k 3 = 4k + 2k 2 + 3k 3 = Since the determinant of the coefficient matrix is nonzero, the system has only the trivial solution. Therefore, the three vectors are linearly independent. (d) By Theorem 5.3.3, any four vectors in R 3 are linearly dependent. 3. (a) Following the technique used in Example 4, we obtain the system of equations 3k + k 2 + 2k 3 + k 4 = 8k + 5k 2 k 3 + 4k 4 = 7k + 3k 2 + 2k 3 = 3k k 2 + 6k 3 + 3k 4 = Since the determinant of the coefficient matrix is nonzero, the system has only the trivial solution. Hence, the four vectors are linearly independent. 3. (b) Again following the technique of Example 4, we obtain the system of equations 3k 2 + k 3 = 3k 2 + k 3 = 2k = 2k k 3 = 24

24 242 Exercise Set 5.3 The third equation, above, implies that k =. This implies that k 3 and hence k 2 must also equal zero. Thus the three vectors are linearly independent. 4. (a) We ask whether there exist constants a, b, and c such that a(2 x + 4x 2 ) + b(3 + 6x + 2x 2 ) + c(2 + x 4x 2 ) = If we equate the coefficients of x, x, and x 2 in the above polynomial to zero, we obtain the following system of equations: 2a + 3b+ 2c = a + 6b+ c = 4a + 2b 4c = Since the coefficient matrix of this system is invertible, the trivial solution is the only solution. Hence, the polynomials are linearly independent. (d) If we set up this problem in the same way we set up Part (a), above, we obtain three equations in four unknowns. Since this is equivalent to having four vectors in R 3, the vectors are linearly dependent by Theorem (a) The vectors lie in the same plane through the origin if and only if they are linearly dependent. Since the determinant of the matrix is not zero, the matrix is invertible and the vectors are linearly independent. Thus they do not lie in the same plane. 6. (a) Since v 2 = 2v, the vectors v and v 2 lie on the same line. But since v 3 is not a multiple of v or v 2, the three vectors do not lie on the same line through the origin. (c) Since v = 2v 2 = 2v 3, these vectors all lie on the same line through the origin. 7. (a) Note that 7v 2v 2 + 3v 3 =.

25 Exercise Set (a) By inspection, we see that (, 2, 3, 4) + (,,, ) = (, 3, 3, 3) so we have the linear combination (, 2, 3, 4) + (,,, ) (, 3, 3, 3) = and this is a linearly dependent set. (b) Using the notation v = (, 2, 3, 4), v 2 = (,,, ), and v 3 = (, 3, 3, 3), the equation from Part (a) becomes v 3 = v + v 2. Solving this for v and then v 2, we have the three dependence relations v = v 2 + v 3 v 2 = v + v 3 v 3 = v + v If there are constants a, b, and c such that a(λ, /2, /2) + b( /2, λ, /2) + c( /2, /2, λ) = (,, ) then λ λ λ a b c = The determinant of the coefficient matrix is 3 3 λ λ = ( λ ) λ This equals zero if and only if λ = or λ = /2. Thus the vectors are linearly dependent for these two values of λ and linearly independent for all other values. 2. Suppose that S has a linearly dependent subset T. Denote its vectors by w,, w m. Then there exist constants k i, not all zero, such that k w + + k m w m = But if we let u,, u n m denote the vectors which are in S but not in T, then k w + + k m w m + u + + u n m =

26 244 Exercise Set 5.3 Thus we have a linear combination of the vectors v,, v n which equals. Since not all of the constants are zero, it follows that S is not a linearly independent set of vectors, contrary to the hypothesis. That is, if S is a linearly independent set, then so is every non-empty subset T. 3. This is similar to Problem. Since {v, v 2,, v r } is a linearly dependent set of vectors, there exist constants c, c 2,, c r not all zero such that c v + c 2 v c r v r = But then c v + c 2 v c r v r + v r+ + + v n = The above equation implies that the vectors v,, v n are linearly dependent. 5. Suppose that {v, v 2, v 3 } is linearly dependent. Then there exist constants a, b, and c not all zero such that (*) av + bv 2 + cv 3 = Case : c =. Then (*) becomes av + bv 2 = where not both a and b are zero. But then {v, v 2 } is linearly dependent, contrary to hypothesis. Case 2: c. Then solving (*) for v 3 yields v 3 = a c v b c v 2 This equation implies that v 3 is in span{v, v 2 }, contrary to hypothesis. Thus, {v, v 2, v 3 } is linearly independent. 6. Note that (u v) + (v w) + (w u) =. 8. Any nonzero vector forms a linearly independent set. The only scalar multiple of a nonzero vector which can equal the zero vector is the zero scalar times the vector.

27 Exercise Set (a) Since sin 2 x + cos 2 x =, we observe that 2(3 sin 2 x) + 3(2 cos 2 x) + ( )(6) = Hence the vectors are linearly dependent. (c) Suppose that there are constants a, b, and c such that a() + b sin x + c sin 2x = Setting x = yields a =. Setting x = π/2 yields b =, and thus, since sin 2x, we must also have c =. Therefore the vectors are linearly independent. (e) Suppose that there are constants a, b, and c such that a(3 x) 2 + b(x 2 6x) + c(5) = or (9a + 5c) + ( 6a 6b)x + (a + b)x 2 = Clearly a = b = (5/9)c. Thus a = 5, b = 5, c = 9 is one solution and the vectors are linearly dependent. This conclusion may also be reached by noting that the determinant of the system of equations is zero. 9a + 5c = 6a 6b = a + b = 2. (a) The Wronskian is x e e e x x x x = e Thus the vectors are linearly independent.

28 246 Exercise Set (b) The Wronskian is sin x cos x xsin x cos x sin x sin x+ xcos x sin x cos x 2cos x xsin x sin x cos x xsin x = cos x sin x sin x+ xcos x 2cos x 2 2 = 2cos x( sin x cos x) = 2cos x Thus the vectors are linearly independent. 23. Use Theorem 5.3., Part (a). 24. (a) False. There are 6 such matrices, namely A = A2 A3 = = A4 = A5 A6 = = Since M 22 has dimension 4, at least two of these matrices must be linear combinations of the other four. In fact, it is easy to show that A, A 2, A 3, and A 4 are linearly independent and that A 5 = A 2 + A 3 + A 4 and A 6 = A + A 3 + A 4. (b) False. One of the vectors might be the zero vector. Otherwise it would be true. (d) False. A finite set of vectors can be linearly dependent without containing the zero vector. 26. We could think of any 3 linearly independent vectors in R 3 as spanning R 3. That is, they could determine directions for 3 (not necessarily orthogonal) coordinate axes. Then any fourth vector would represent a point in this coordinate system and hence be a linear combination of the other 3 vectors.

29 EXERCISE SET (a) This set is a basis. It has two vectors and neither is a multiple of the other. (c) This set is not a basis since one vector is a multiple of the other. 3. (a) This set has the correct number of vectors and they are linearly independent because = 6 Hence, the set is a basis. (c) The vectors in this set are linearly dependent because = Hence, the set is not a basis. 4. (a) The vectors in this set are linearly dependent because = Thus, the set is not a basis. (Compare with Problem 3(c), above.) 247

30 248 Exercise Set (c) This set has the correct number of vectors and = Hence, the vectors are linearly independent and therefore are a basis. 5. The set has the correct number of vectors. To show that they are linearly independent, we consider the equation a b c = d 2 If we add matrices and equate corresponding entries, we obtain the following system of equations: 3a + d = 6a b 8c = 3a b 2c d = 6a 4c + 2d = Since the determinant of the coefficient matrix is nonzero, the system of equations has only the trivial solution; hence, the vectors are linearly independent. 6. (a) Recall that cos 2x = cos 2 x sin 2 x; that is, v + ( )v 2 + ( )v 3 = Hence, S is not a linearly independent set of vectors. (b) We can use the above identity to write any one of the vectors v i as a linear combination of the other two. Since no one of these vectors is a multiple of any other, they are pairwise linearly independent. Thus any two of these vectors form a basis for V. 7. (a) Clearly w = 3u 7u 2, so the coordinate vector relative to {u, u 2 } is (3, 7).

31 Exercise Set (b) If w = au + bu 2, then equating coordinates yields the system of equations 2a + 3b = 4a + 8b = This system has the solution a = 5/28, b = 3/4. Thus the desired coordinate vector is (5/28, 3/4). 8. (a) Since w = 2 (u + u 2 ), the coordinate vector of w relative to S = {u, u 2 } is (w) S = ( 2, 2 ). (b) Since w = 2 ( u + u 2 ), the coordinate vector of w relative to S = {u, u 2 } is (w) S = ( 2, 2 ). (c) Since w = u 2 = u + u 2, the coordinate vector of w relative to S = {u, u 2 } is (w) S = (, ). 9. (a) If v = av + bv 2 + cv 3, then a + 2b + 3c = 2 2b + 3c = 3c =3 From the third equation, c =. Plugging this value into the second equation yields b = 2, and finally, the first equation yields a = 3. Thus the desired coordinate vector is (3, 2, ).. (b) If p = ap + bp 2 + cp 3, then equating coefficients yields the system of equations a + b = 2 a + c = b + c = This system has the solution a =, b = 2, and c =. Thus the desired coordinate vector is (, 2, ). 2. The augmented matrix of the system reduces to Hence, x = s, x 2 =, and x 3 = s. Thus the solution space is spanned by (,, ) and has dimension.

32 25 Exercise Set If we reduce the augmented matrix to row-echelon form, we obtain 3 Thus x = 3r s, x 2 = r, and x 3 = s, and the solution vector is x x x 2 3 = 3r s 3 r = s r+ s Since (3,, ) and (,, ) are linearly independent, they form a basis for the solution space and the dimension of the solution space is Since the determinant of the system is not zero, the only solution is x = x 2 = x 3 =. Hence there is no basis for the solution space and its dimension is zero. 9. (a) Any two linearly independent vectors in the plane form a basis. For instance, (,, ) and (, 5, 2) are a basis because they satisfy the plane equation and neither is a multiple of the other. (c) Any nonzero vector which lies on the line forms a basis. For instance, (2,, 4) will work, as will any nonzero multiple of this vector. (d) The vectors (,, ) and (,, ) form a basis because they are linearly independent and a(,, ) + c(,, ) = (a, a + c, c) 2. This space is spanned by the vectors, x, x 2 and x 3. Only the last three vectors form a linearly independent triple. Thus the space has dimension (a) We consider the three linear systems k + k = 2 2k 2k = 2 3k 2k = 2

33 Exercise Set which give rise to the matrix A row-echelon form of the matrix is 3 2 from which we conclude that e 3 is in the span of {v, v 2 }, but e and e 2 are not. Thus {v, v 2, e } and {v, v 2, e 2 } are both bases for R We consider the linear system of equations k v + k 2 v 2 + k 3 e + k 4 e 2 + k 5 e 3 + k 6 e 4 = which gives rise to the coefficient matrix A row-echelon form of the matrix is If we eliminate any two of the final four columns of this matrix, we obtain a 4 4 matrix. If the determinant of this matrix is zero, then the system of equations has a nontrivial solution, so the corresponding vectors are linearly dependent. Otherwise, they are linearly independent. If we include column 3 (which corresponds to e ) in the determinant, its value is zero. Otherwise, it is not. Thus we may add any two of the vectors e 2, e 3, and e 4 to the set {v, v 2 } to obtain a basis for R 4.

34 252 Exercise Set Since {u, u 2, u 3 } has the correct number of vectors, we need only show that they are linearly independent. Let au + bu 2 + cu 3 = Thus av + b(v + v 2 ) + c(v + v 2 + v 3 ) = or (a + b + c)v + (b + c)v 2 + cv 3 = Since {v, v 2, v 3 } is a linearly independent set, the above equation implies that a + b + c = b + c = c =. Thus, a = b = c = and {u, u 2, u 3 } is also linearly independent. 24. (a) Note that the polynomials, x, x 2,, x n form a set of n + linearly independent functions in F(, ). (b) From Part (a), dim F(, ) > n for any positive integer n. Thus F(, ) is infinitedimensional. 25. First notice that if v and w are vectors in V and a and b are scalars, then (av + bw) S = a(v) S + b(w) S. This follows from the definition of coordinate vectors. Clearly, this result applies to any finite sum of vectors. Also notice that if (v) S = () S, then v =. Why? Now suppose that k v + + k r v r =. Then (k v + + k r v r ) S = k (v ) S + + k r (v r ) S = () S Conversely, if k (v ) S + + k r (v r ) S, = () S, then (k v + + k r v r ) S = () S, or k v + + k r v r = Thus the vectors v,, v r are linearly independent in V if and only if the coordinate vectors (v ) S,, (v r ) S are linearly independent in R n.

35 Exercise Set If every vector v in V can be written as a linear combination v = a v + + a r v r of v,, v r, then, as in Exercise 24, we have (v) S = a (v ) S + + a r (v r ) S. Hence, the vectors (v ) S,, (v r ) S span a subspace of R n. But since V is an n-dimensional space with, say, the basis S = {u,, u n }, then if u = b u + + b n u n, we have (u) S = (b,, b n ); that is, every vector in R n represents a vector in V. Hence {(v ) S,, (v r ) S } spans R n. Conversely, if {(v ) S,, (v r ) S } spans R n, then for every vector (b,, b n ) in R n, there is an r-tuple (a,, a r ) such that (b,, b n ) = a (v ) S + + a r (v r ) S = (a v + + a r v r ) S Thus a v + + a r v r = b u + + b n u n, so that every vector in V can be represented as a linear combination of v,, v r. 27. (a) Let v, v 2, and v 3 denote the vectors. Since S = {, x, x 2 } is the standard basis for P 2, we have (v ) S = (,, 2), (v 2 ) S = (3, 3, 6), and (v 3 ) S = (9,, ). Since {(,, 2), (3, 3, 6), (9,, )} is a linearly independent set of three vectors in R 3, then it spans R 3. Thus, by Exercises 24 and 25, {v, v 2, v 3 } is linearly independent and spans P 2. Hence it is a basis for P (a) It is clear from the picture that the x -y coordinates of (, ) are (, 2). y l y' 2 45º l x and x' (d) Let (a, b) and (a, b ) denote the coordinates of a point with respect to the x-y and x - y coordinate systems, respectively. If (a, b) is positioned as in Figure (), then we have a = a b and b = 2b

36 254 Exercise Set (d) y y y' (a, b) (a', b') (a, b) (a', b') y' a' 45 a 2b 45 b b x and x' 2b 45 a b a 45 x and x' Figure Figure 2 These formulas hold no matter where (a,b) lies in relation to the coordinate axes. Figure (2) shows another configuration, and you should draw similar pictures for all of the remaining cases. 3. See Theorem There is. Consider, for instance, the set of matrices A = B = C = D and = Each of these matrices is clearly invertible. To show that they are linearly independent, consider the equation aa + bb + cc + dd = This implies that a b c d =

37 Exercise Set The above 4 4 matrix is invertible, and hence a = b = c = d = is the only solution. And since the set {A, B, C, and D} consists of 4 linearly independent vectors, it forms a basis for M 22. a c 32. (b) The most general 2 2 symmetric matrix has the form. Hence, if n = 2, the matrices c b,, and form a basis and the dimension is therefore 3. If n = 3, the matrices,,,,and form a basis and the dimension is therefore 6. In general, there are n 2 n + n n( n+ ) elements in an n n matrix with = on or 2 2 above the main diagonal. Therefore the dimension of the subspace of n n symmetric matrices is n(n + )/2. (c) Since there are n 2 elements in each n n matrix, with n elements on the main diagonal, there are (n 2 n)/2 elements above the diagonal. Thus, any triangular matrix can have at most (n 2 n)/2 + n = (n 2 + n)/2 nonzero elements. The set of n n matrices consisting of all zeros except for a in each of these spots will form a basis for the space. Consequently, the space will have dimension n(n + )/ (a) The set has elements in a 9 dimensional space.

38 35. (b) The equation x + x x n = can be written as x = x 2 x 3 x n where x 2, x 3,, x n can all be assigned arbitrary values. Thus, its solution space should have dimension n. To see this, we can write The n vectors in the above equation are linearly independent, so the vectors do form a basis for the solution space. 36. (a) If p () = p 2 () =, then (p + p 2 )() = p () + p 2 () =. Also, kp () =. Hence W is closed under both vector addition and scalar multiplication. (b) We are looking at the subspace W of polynomials ax 2 + bx + c in P 2 where a + b + c = or c = (a + b). Thus it would appear that the dimension of W is 2. (c) The polynomial p(x) = ax 2 + bx + c will be in the subspace W if and only if p() = a + b + c =, or c = a b. Thus, where a and b are arbitrary. The vectors and are linearly independent and hence the polynomials x 2 and x form a basis for W. a b c a b a b a = = + b = + x x x n x x x x x x x n n = x x 2 3 x n 256 Exercise Set 5.4

39 EXERCISE SET (b) Since the equation Ax = b has no solution, b is not in the column space of A. (c) Since A = b, we have b = c 3c 2 + c 3. (d) Since A = b, we have b = c + (t )c 2 + tc 3 for all real numbers t. 5. (a) The general solution is x = + 3t, x 2 = t. Its vector form is Thus the vector form of the general solution to Ax = is (c) The general solution is x = + 2r s 2t, x 2 = r, x 3 = s, x 4 = t. Its vector form is r s 2 + t t t t t 3 257

40 Thus the vector form of the general solution to Ax = is 6. (a) Since the reduced row-echelon form of A is the solution to the equation Ax = is x = 6t, x 2 = 9t, x 3 = t. Thus is a basis for the nullspace. (c) Since the reduced row-echelon form of A is one solution to the equation Ax = is x = s + 2t, x 2 = s 4t, x 3 = s, x 4 = 7t. Thus the set of vectors is a basis for the nullspace and r s t Exercise Set 5.5

41 (e) The reduced row-echelon form of A is One solution to the equation Ax = is x = 2s 6t, x 2 = 2t, x 3 = 5t, x 4 = s, and x 5 = 2t. Hence the set of vectors is a basis for the nullspace of A. 8. (a) From a row-echelon form of A, we have that the vectors (,, 3) and (,, 9) are a basis for the row space of A. (c) From a row-echelon form of A, we have that the vectors (, 4, 5, 2) and (,,, 4/7) are a basis for the row space of A. 9. (a) One row-echelon form of A T is Thus a basis for the column space of A is 5 7 and and Exercise Set

42 26 Exercise Set (c) One row-echelon form of A T is 2 Thus a basis for the column space of A is 2 and. (a) Since, by 8(a), the row space of A has dimension 2, any two linearly independent row vectors of A will form a basis for the row space. Because no row of A is a multiple of another, any two rows will do. In particular, the first two rows form a basis and Row 3 = Row 2 + 2(Row ). (c) Refer to 8(c) and use the solution to (a), above. In particular, the first two rows form a basis and Row 3 = Row Row 2. (e) Let r, r 2,, r 5 denote the rows of A. If we observe, for instance, that r = r 3 + r 4 and that r 2 = 2r + r 5, then we see that {r 3, r 4, r 5 } spans the row space. Since the dimension of this space is 3 (see the solution to Exercise 6(e)), the set forms a basis. For those who don t wish to rely on insight, set ar + br 2 + cr 3 + dr 4 + er 5 = and solve the resulting homogeneous system of equations by finding the reduced rowechelon form for A T. This yields a = s + 2t b = t c = s d = s e = t so that ( s + 2t)r tr 2 sr 3 + sr 4 + tr 5 =, or s( r r 3 + r 4 ) + t(2r r 2 + r 5 ) =

43 Exercise Set Since this equation must hold for all values of s and t, we have r = r 3 + r 4 and r 2 = 2r + r 5 which is the result obtained above.. (a) The space spanned by these vectors is the row space of the matrix One row-echelon form of the above matrix is and the reduced row-echelon form is Thus {(,, 4, 3), (,, 5, 2), (,,, /2)} is one basis. Another basis is {(,,, /2), (,,, 9/2), (,,, /2)}. 2. (a) If we solve the vector equation (*) av + bv 2 + cv 3 + dv 4 = we obtain the homogeneous system a 3b c 5d = 3b + 3c + 3d = a + 7b + 9c + 5d = a + b + 3c d =

44 262 Exercise Set 5.5 The reduced row-echelon form of the augmented matrix is 2 2 Thus {v, v 2 } forms a basis for the space. The solution is a = 2s + 2t, b = s t, c = s, d = t. This yields ( 2s + 2t)v + ( s t)v 2 + sv 3 + tv 4 = or s( 2v v 2 + v 3 ) + t(2v v 2 + v 4 ) = Since s and t are arbitrary, set s =, t = and then s =, t = to obtain the dependency equations 2v v 2 + v 3 = 2v v 2 + v 4 = Thus v 3 = 2v + v 2 and v 4 = 2v + v 2 2. (c) If we solve the vector equation (*) av + bv 2 + cv 3 + dv 4 + ev 5 = we obtain the homogeneous system a 2b + 4c 7e = a + 3b 5c + 4d+ 8e = 5a + b + 9c + 2d + 2e = 2a + 4c 3d 8e =

45 Exercise Set The reduced row-echelon form of the augmented matrix is This tells us that {v, v 2, v 4 } is the desired basis. The solution is a = 2s + t, b = s 3t, c = s, d = 2t, and e = t. This yields ( 2s + t)v + (s 3t)v 2 + sv 3 2tv 4 + tv 5 = or s( 2v + v 2 + v 3 ) + t(v 3v 2 2v 4 + v 5 ) = Since s and t are arbitrary, set s =, t = and then s =, t = to obtain the dependency equations 2v + v 2 + v 3 = v 3v 2 2v 4 + v 5 = Thus v 3 = 2v v 2 and v 5 = v + 3v 2 + 2v 4 3. Let A be an n n invertible matrix. Since A T is also invertible, it is row equivalent to I n. It is clear that the column vectors of I n are linearly independent. Hence, by virtue of Theorem 5.5.5, the column vectors of A T, which are just the row vectors of A, are also linearly independent. Therefore the rows of A form a set of n linearly independent vectors in R n, and consequently form a basis for R n. 5. (a) We are looking for a matrix so that the only solution to the equation Ax = is x =. Any invertible matrix will satisfy this condition. For example, the nullspace of the matrix A = is the single point (,, ).

46 (b) In this case, we are looking for a matrix so that the solution of Ax = is one-dimensional. Thus, the reduced row-echelon form of A has one column without a leading one. As an example, the nullspace of the matrix A = is span, a line in R 3. (c) In this case, we are looking for a matrix so that the solution space of Ax = is two-dimensional. Thus, the reduced row-echelon form of A has two columns without leading ones. As an example, the nullspace of the matrix A = is span, a plane in R (a) True. Since premultiplication by an elementary matrix is equivalent to an elementary row operation, the result follows from Theorem (c) False. For instance, let A = and EA =. Then the column space of A is span and the column space of EA is span. These are not the same spaces. (d) True by Theorem 5.5. (e) False. The row space of an invertible n n matrix is the same as the row space of I n, which is R n. The nullspace of an invertible matrix is just the zero vector , 264 Exercise Set 5.5

47 Exercise Set s 5s (a) The matrices will all have the form s t where s and t are any real numbers. 3t 5t = (b) Since A and B are invertible, their nullspaces are the origin. The nullspace of C is the line 3x + y =. The nullspace of D is the entire xy-plane. 8. Let A = [ ]. Then Ax = [] has the particular solution [ ] T and Ax = has the general solution [s t s t] T. Thus the general solution can be written as x x x 2 3 = + s + t 9. Theorem: If A and B are n n matrices and A is invertible, then the row space of AB is the row space of B. Proof: If A is invertible, then there exist elementary matrices E, E 2,, E k such that A = E E 2 E k I n or AB = E E 2 E k B Thus, Theorem guarantees that AB and B will have the same row spaces.

48

49 EXERCISE SET (a) The reduced row-echelon form for A is 6 9 Thus rank (A) = 2. The solution to Ax = is x = 6t, y = 9t, z = t, so that the nullity is one. There are three columns, so we have 2 + = 3. (c) The reduced row-echelon form for A is Thus rank (A) = 2. The null space will have dimension two since the solution to Ax = has two parameters. There are four columns, so we have = Recall that rank(a) is the dimension of both the row and column spaces of A. Use the Dimension Theorem to find the dimensions of the nullspace of A and of A T, recalling that if A is m n, then A T is n m, or just refer to the chart in the text. 7. Use Theorems and (a) The system is consistent because the two ranks are equal. Since n = r = 3, n r = and therefore the number of parameters is. (b) The system is inconsistent because the two ranks are not equal. 267

50 268 Exercise Set (d) The system is consistent because the two ranks are equal. Here n = 9 and r = 2, so that n r = 7 parameters will appear in the solution. (f) Since the ranks are equal, the system is consistent. However A must be the zero matrix, so the system gives no information at all about its solution. This is reflected in the fact that n r = 4 = 4, so that there will be 4 parameters in the solution for the 4 variables. 9. The system is of the form Ax = b where rank(a) = 2. Therefore it will be consistent if and only if rank([a b]) = 2. Since [A b] reduces to 3 b b b b 4b + 3b b + b 2b b5 8b 2 + b the system will be consistent if and only if b 3 = 4b 2 3b, b 4 = b 2 + 2b, and b 5 = 8b 2 7b, where b and b 2 can assume any values.. Suppose that A has rank 2. Then two of its column vectors are linearly independent. Thus, by Theorem 5.6.9, at least one of the 2 2 submatrices has nonzero determinant. Conversely, if at least one of the determinants of the 2 2 submatrices is nonzero, then, by Theorem 5.6.9, at least two of the column vectors must be linearly independent. Thus the rank of A must be at least 2. But since the dimension of the row space of A is at most 2, A has rank at most 2. Thus, the rank of A is exactly 2.. If the nullspace of A is a line through the origin, then it has the form x = at, y = bt, z = ct where t is the only parameter. Thus nullity(a) = 3 rank(a) =. That is, the row and column spaces of A have dimension 2, so neither space can be a line. Why?

51 Exercise Set (a) If we attempt to reduce A to row-echelon form, we find that t t A t t t t if t t t t 2 t if t ( + t) t t if t 2 t + 2 Thus rank(a) = 3 if t,, 2. If t =, rank(a) = 3 by direct computation. If t =, rank(a) = by inspection, and if t = 2, rank(a) = 2 by the above reduction. 3. Call the matrix A. If r = 2 and s =, then clearly rank(a) = 2. Otherwise, either r 2 or s and rank(a) = 3. Rank(A) can never be. 4. Call the matrix A and note that rank(a) is either or 2. Why? By Exercise, rank(a) = if and only if x y x x z y z =, =, and = y x y Thus we must have x 2 y = xy z = y 2 xz =. If we let x = t, the result follows. 6. (a) The column space of the matrix is the xy-plane.

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