Page 6 of 7 4. (12 points) Math 2333: THQ 09, 23-03-17, 18:19 Indicate which of the following collections of objects form a vector space with the given operations. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. It is not sufficient to say that this is dierent from the vectors you already know; it would have to break one (or more) of the rules above. (If desired you may use the definition in the book, but that would require more eort.) (a) All 3 5 matrices with the usual matrix addition and scalar multiplication. Yes, or No, because: (b) R3 with the usual addition and scalar multiplication. Yes, or No, because: (c) All polynomial functions f (x) of degree three (cubic polynomials), with the usual addition and scaling for polynomials. Yes, or No, because: (d) A tiny space with a single object 4 with addition defined by 4 + 4 = 4 and scalar multiplication defined by k 4 = 4 for any real number k. Yes, or No, because: (e) The real numbers R with arithmetic addition and multiplication for its vector addition and scaling. Yes, or No, because: (f) All pairs (x, y) of real numbers x and y with the expected scalar multiplication k(a, b) = (ka, kb) but a strange addition defined by (a, b) addition as a reminder that it is weird. Yes, or No, because: (c, d) = (a + d, b + c). We use the symbol for this vector Page 7 of 7 Math 2333: THQ 09, 23-03-17, 18:19 5. (12 points) Indicate which of the following collections of objects for a vector space with the given operations. Here you may assume that the parent space is a subspace, so you only have to check the closure conditions to see if you have a valid subspace. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. (a) All symmetric matrices, as part of the space M3,3 of 3 3 matrices with the usual operations. Yes, or No, because: (b) All upper triangular and lower triangular matrices from M2,2 , of 2 2 matrices. Yes, or No, because: (c) All polynomials f (x) of degree at most 4 that satisfy f (2) = 5. Yes, or No, because: (d) All polynomials f (x) of degree at most 4 that satisfy f (3) = 0. Yes, or No, because: (e) All upper triangular matrices from M3,3 , of 3 3 matrices. Yes, or No, because: (f) All matrices A from M3,3 , of 3 3 matrices with the property that tr A = 0 (zero trace). Yes, or No, because: Page 6 of 7 4. (12 points) Math 2333: THQ 09, 23-03-17, 18:19 Indicate which of the following collections of objects form a vector space with the given operations. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. It is not sufficient to say that this is dierent from the vectors you already know; it would have to break one (or more) of the rules above. (If desired you may use the definition in the book, but that would require more eort.) (a) All 3 5 matrices with the usual matrix addition and scalar multiplication. Yes, or No, because: (b) R3 with the usual addition and scalar multiplication. Yes, or No, because: (c) All polynomial functions f (x) of degree three (cubic polynomials), with the usual addition and scaling for polynomials. Yes, or No, because: (d) A tiny space with a single object 4 with addition defined by 4 + 4 = 4 and scalar multiplication defined by k 4 = 4 for any real number k. Yes, or No, because: (e) The real numbers R with arithmetic addition and multiplication for its vector addition and scaling. Yes, or No, because: (f) All pairs (x, y) of real numbers x and y with the expected scalar multiplication k(a, b) = (ka, kb) but a strange addition defined by (a, b) addition as a reminder that it is weird. Yes, or No, because: (c, d) = (a + d, b + c). We use the symbol for this vector Page 7 of 7 Math 2333: THQ 09, 23-03-17, 18:19 5. (12 points) Indicate which of the following collections of objects for a vector space with the given operations. Here you may assume that the parent space is a subspace, so you only have to check the closure conditions to see if you have a valid subspace. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. (a) All symmetric matrices, as part of the space M3,3 of 3 3 matrices with the usual operations. Yes, or No, because: (b) All upper triangular and lower triangular matrices from M2,2 , of 2 2 matrices. Yes, or No, because: (c) All polynomials f (x) of degree at most 4 that satisfy f (2) = 5. Yes, or No, because: (d) All polynomials f (x) of degree at most 4 that satisfy f (3) = 0. Yes, or No, because: (e) All upper triangular matrices from M3,3 , of 3 3 matrices. Yes, or No, because: (f) All matrices A from M3,3 , of 3 3 matrices with the property that tr A = 0 (zero trace). Yes, or No, because: Q4a) Yes b) yes c) No for example f = x3 + 3x g(x) = -x3 f+g = 3x, which is not the cubic polynomial therefore, f+g v d) yes e) yes f) No because addition is not commutative for example (1,2) + (3,5) = (1+5, 2+3) = (6,5) (3,5) + (1,2) = (3+2, 5+1) = (5,6) (a,b) + (c,d) ( c , d ) +(a , b) Q5a) Yes. Because sum of any two symmetric matrices is also symmetric matrix b) Yes because sum of any two upper triangular matrices is also an upper triangular matrix Sum of any two lower triangular matrices is alos a lower triangular matrix There exist a linear transformation for |R2*2 to |R2*2 whose kernel consist of all lower triangular 2*2 matrices, while the image consist of all upper triangular 2*2 matrice c) No Let, f be in this set So, f(2)=5 (2f)(2)=2f(2)=10 Hence, 2f is not in the set ie set if not closed under scalar multiplication d) Let, f ,g polynomials which satisfy (f + g)(0)=f(0)+g(0)=0 Hence, f + g is in the set Hence a subspace e) Yes f) Yes Page 6 of 7 4. (12 points) Math 2333: THQ 09, 23-03-17, 18:19 Indicate which of the following collections of objects form a vector space with the given operations. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. It is not sufficient to say that this is dierent from the vectors you already know; it would have to break one (or more) of the rules above. (If desired you may use the definition in the book, but that would require more eort.) (a) All 3 5 matrices with the usual matrix addition and scalar multiplication. Yes, or No, because: (b) R3 with the usual addition and scalar multiplication. Yes, or No, because: (c) All polynomial functions f (x) of degree three (cubic polynomials), with the usual addition and scaling for polynomials. Yes, or No, because: (d) A tiny space with a single object 4 with addition defined by 4 + 4 = 4 and scalar multiplication defined by k 4 = 4 for any real number k. Yes, or No, because: (e) The real numbers R with arithmetic addition and multiplication for its vector addition and scaling. Yes, or No, because: (f) All pairs (x, y) of real numbers x and y with the expected scalar multiplication k(a, b) = (ka, kb) but a strange addition defined by (a, b) addition as a reminder that it is weird. Yes, or No, because: (c, d) = (a + d, b + c). We use the symbol for this vector Page 7 of 7 Math 2333: THQ 09, 23-03-17, 18:19 5. (12 points) Indicate which of the following collections of objects for a vector space with the given operations. Here you may assume that the parent space is a subspace, so you only have to check the closure conditions to see if you have a valid subspace. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail. (a) All symmetric matrices, as part of the space M3,3 of 3 3 matrices with the usual operations. Yes, or No, because: (b) All upper triangular and lower triangular matrices from M2,2 , of 2 2 matrices. Yes, or No, because: (c) All polynomials f (x) of degree at most 4 that satisfy f (2) = 5. Yes, or No, because: (d) All polynomials f (x) of degree at most 4 that satisfy f (3) = 0. Yes, or No, because: (e) All upper triangular matrices from M3,3 , of 3 3 matrices. Yes, or No, because: (f) All matrices A from M3,3 , of 3 3 matrices with the property that tr A = 0 (zero trace). Yes, or No, because: Q4a) Yes b) yes c) No for example f = x3 + 3x g(x) = -x3 f+g = 3x, which is not the cubic polynomial therefore, f+g v d) yes e) yes f) No because addition is not commutative for example (1,2) + (3,5) = (1+5, 2+3) = (6,5) (3,5) + (1,2) = (3+2, 5+1) = (5,6) (a,b) + (c,d) ( c , d ) +(a , b) Q5a) Yes. Because sum of any two symmetric matrices is also symmetric matrix b) Yes because sum of any two upper triangular matrices is also an upper triangular matrix Sum of any two lower triangular matrices is alos a lower triangular matrix There exist a linear transformation for |R2*2 to |R2*2 whose kernel consist of all lower triangular 2*2 matrices, while the image consist of all upper triangular 2*2 matrice c) No Let, f be in this set So, f(2)=5 (2f)(2)=2f(2)=10 Hence, 2f is not in the set ie set if not closed under scalar multiplication d) Let, f ,g polynomials which satisfy (f + g)(0)=f(0)+g(0)=0 Hence, f + g is in the set Hence a subspace e) Yes f) Yes