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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 25",
      "body": "Let S be the set of all vectors **v** such that **v** = < x , y, z > with 3x + 4y = 1 and z = 0. Then, x = https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png and y = https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png. So, **v** = <https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png, https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png, 0>\n\nProof:\n\nAssume S is a real linear space, then S must be closed under addition (axiom 1). In other words, if **v** and **w** are in S, **v+w** must also be in S. By that logic, if **v** is in S, **v + v** = 2**v** must also be in S. Another way of saying this is that S must be closed under scalar multiplication (axiom 2).\n\nBut for all **v** in S, **v** must equal <https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png, https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png, 0>\n\nYet multiplying **v** by any constant _c_, breaks that form. So S cannot possibly be closed under scalar multiplication or addition.\n\nTherefore, S cannot be a real linear space.\n\nQED",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 25",
      "body": "Let S be the set of all vectors **v** such that **v** = < x , y, z > with 3x + 4y = 1 and z = 0. Then, x = https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png and y = https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png. So, **v** = <https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png, https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png, 0>\n\nProof:\n\nAssume S is a real linear space, then S must be closed under addition (axiom 1). In other words, if **v** and **w** are in S, **v+w** must also be in S. By that logic, if **v** is in S, **v + v** = 2**v** must also be in S. Another way of saying this is that S must be closed under scalar multiplication (axiom 2).\n\nBut for all **v** in S, **v** must equal <https://quicklatex.com/cache3/97/ql_9ee79f4b5cf0fc753d91e15b36107097_l3.png, https://quicklatex.com/cache3/f3/ql_fe6c307414aa004c4f093d48e79ba3f3_l3.png, 0>\n\nYet multiplying **v** by any constant _c_, breaks that form. So S cannot possibly be closed under scalar multiplication or addition.\n\nTherefore, S cannot be a real linear space.\n\nQED",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 23",
      "body": "Let S be the set of all vectors **v** = <x, y, z> in V<sub>3</sub> with either x = 0 or y = 0. Assume S is a real linear space. Then, S must be closed under addition (axiom 1). Let **v<sub>1</sub>** = <0, b, c> &#x2208; S and let **v<sub>2</sub>** = <a, 0, c> &#x2208; S, where a, b, and c are non-zero. Since S is a real linear space, **v<sub>3</sub>** = **v<sub>1</sub>**+**v<sub>2</sub>** must be in S. But **v<sub>3</sub>** = <a, b, 2c>, which has neither a zero x-component nor a zero y-component, since a, b, and c are non-zero. So **v<sub>3</sub>** cannot be in S. Thus, S is not closed under addition. Therefore, S cannot possibly be a real linear space.\n\nQED",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 11",
      "body": "Let S be the set of all functions &#x192; such that for all x in their domain, the function is increasing. In other words,\nS = { &#x192; | &#x192;' > 0 &#x2200;&#x192; }\n\nAssume that S is a real linear space. Then S will be closed under scalar multiplication. So, for all functions &#x192; &#x2208; S, and all real numbers _a_, _a_&#x192; &#x2208; S also. We will show that there are some real numbers _a_ for which _a_&#x192; &#x2209; S, even though &#x192; &#x2208; S.\n\nProof:\n\nAssume &#x192; &#x2208; S. Let _a_ &#x2208; R such that _a_ < 0. Then (_a_&#x192;)' = _a_(&#x192;)' = _a_&#x192;'. Then _a_&#x192;' will be less than zero. Therefore, _a_&#x192; &#x2209; S (&#x2200;&#x192; &#x2208; S). So, S cannot possibly be a real linear space.\n\nQED",
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      "title": "Compilation of Solutions to Tom Apostol's Infamously Rigourous Calculus Textbook (vol. 2)",
      "body": "Hello!\n\nSomeone else has already compiled solutions to all exercises in Tom Apostol's \"Calculus\", but on for vol. 1. As I am taking multivariable calculus this semester, I've decided it would be beneficial (as a study tool) to publish my solutions online to the second volume. Apostol provides basic solutions, but I thought I'd expand on his.\n\nThis is a very big project. I expect it to take about a year. I will be updated my Steemit page as I get further and further along.\n\nHere a few links:\n\n#### Vol. 1 solutions: (single variable calc)\n\n[RoRi's Solutions to Volume 1 on his blog Stumbling Robot](http://www.stumblingrobot.com/index-of-solutions/solutions-to-calculus-exercises/) (opens different site)\n\n#### Vol. 2 solutions: (multivariable calc)\n\n[My ongoing solutions to exercises 1.5](https://steemit.com/apostol-calculus/@shinday/apostol-s-calculus-exercises-1-5) (opens a steemit page)\n\n[My \"master\" page for all solutions, very much incomplete](https://steemit.com/mathematics/@shinday/tom-apostol-s-calculus-vol-2-my-solutions-to-exercises) (steemit page)\n\nAlthough my solutions are very incomplete so far, I would love criticism on my proofs that I have. You can find everything from these links. If you need help getting access to the PDFs for these books, send me a message.",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 14",
      "body": "Let S be the set of all functions f such that https://quicklatex.com/cache3/8a/ql_f82f26c1a83fe7f85f466da161a75f8a_l3.png. Assume that S is a real linear space. Then, by the addition axioms of real linear spaces, there must exist an element g in S such that f + g = 0. We will show that this leads t a contradiction, thus proving S cannot be a real linear space.\n\nAssume f is in S, and has the additive inverse (axiom 6) g such that f + g = 0. Then the primitive of f, F(x) satisfies the property:\n\nF(1) - F(0) ≥ 0.\n\nThis implies that F(1) ≥ F(0), so f(x) must be greater than zero, since f(x) is the derivative of F(x), and F(x) must be increasing. \n\nAt the very least, (f(1) + f(0))/2 ≥ 0 (the average slope of the primitive must be positive or zero)\n\nBut this must also hold true for g and its primitive G, since g is also an element of S. But if f ≥ 0 and g ≥ 0, then (f + g) ≥ 0. The only (f + g) = 0 is if f and g are both zero, since neither can be less than zero.\n\nBut S allows for all functions such that https://quicklatex.com/cache3/8a/ql_f82f26c1a83fe7f85f466da161a75f8a_l3.png.\n\nThus, there cannot exist a negative element g for every element f in S for https://quicklatex.com/cache3/8a/ql_f82f26c1a83fe7f85f466da161a75f8a_l3.png. Only those functions f that are equal to zero can have a negative element. Therefore, S cannot be a real linear space.\n\nQED",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 11",
      "body": "Let S be the set of all functions such that for all x in their domain, the function is increasing. In other words,\nS = { f(x) | f'(x) > 0 ∀x }\nAssume that S is a real linear space. Then S will be closed under multiplication. So, for all functions f ∈ S and g ∈ S, f⋅g will also be in S. We will show that that there exist functions such that this is not true.\n\nProof:\nAssume f and g are both in S. Then f⋅g will also be in S. Since S is the set of all functions for which their derivative is always positive, then the following three conditions must hold if S is to be a real linear space:\n\nf'(x) > 0 ∀x \ng'(x) > 0 ∀x \n(f⋅g)'(x) > 0 ∀x\n\nBy a simple application of the product rule, we can deduce that:\n\n(f⋅g)'(x) = (f'⋅g + f⋅g')(x)\n\nNotice that if f and g are either both negative, or just one is negative with a greater absolute value than the other, then the derivative of their product is not necessarily greater than 0. Thus, S cannot be closed under multiplication.\n\nJustification: It is only required that the derivative be greater than 0; there is no restriction on the actual function value, except that it must be increasing with respect to x.\n\nIt follows that S cannot possibly be a real linear space.\n\nQED",
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      "permlink": "apostol-s-calculus-exercises-1-5",
      "title": "[Apostol's Calculus] Exercises 1.5",
      "body": "The key to solving these exercises is to analyze each of the given sets and determine whether or not they violate any of the 10 axioms of real linear vectors spaces V<sub>n</sub>:\n1. Closure under addition\n2. Closure under multiplication\n3. Commutativity of addition\n4. Associativity of Addition\n5. Existence of 0 element\n6. Existence of negatives\n7. associativity of scalar multiplication\n8. Distributive law for addition in V\n9. Distributive law for addition of numbers\n10. Existence of Identity Element\n\nFor these solutions, I will simply state \"Yes\" if the example given is a real linear space, but I will provide a formal proof if it is not.\n\nAnswers:\n1. Yes\n2. Yes\n3. Yes\n4. Yes\n5. No ([see proof](https://steemit.com/mathematics/@shinday/apostol-s-calculus-exercises-1-5-solution-to-5))\n6. Yes\n7. Yes\n8. Yes\n9. Yes\n10. Yes\n11. No (see proof)\n12. Yes\n13. Yes\n14. No (see proof)\n15. Yes\n16. Yes\n17. Yes\n18. Yes\n19. Yes\n20. Yes\n21. Yes\n22. Yes\n23. No (see proof)\n24. Yes\n25. No (see proof)\n26. Yes\n27. Yes\n28. Yes\n29.\n30.\n31.\n     (a) No\n     (b) No\n     (c) No\n     (d) No\n32.",
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      "permlink": "apostol-s-calculus-exercises-1-5",
      "title": "[Apostol's Calculus] Exercises 1.5",
      "body": "The key to solving these exercises is to analyze each of the given sets and determine whether or not they violate any of the 10 axioms of real linear vectors spaces V<sub>n</sub>:\n1. Closure under addition\n2. Closure under multiplication\n3. Commutativity of addition\n4. Associativity of Addition\n5. Existence of 0 element\n6. Existence of negatives\n7. associativity of scalar multiplication\n8. Distributive law for addition in V\n9. Distributive law for addition of numbers\n10. Existence of Identity Element\n\nFor these solutions, I will simply state \"Yes\" if the example given is a real linear space, but I will provide a formal proof if it is not.\n\nAnswers:\n1. Yes\n2. Yes\n3. Yes\n4. Yes\n5. No ([see proof](https://steemit.com/mathematics/@shinday/apostol-s-calculus-exercises-1-5-solution-to-5))\n6. Yes\n7. Yes\n8. Yes\n9. Yes\n10. Yes\n11. No (see proof)\n12. Yes\n13. Yes\n14. No (see proof)\n15. Yes\n16. Yes\n17. Yes\n18. Yes\n19. Yes\n20. Yes\n21. Yes\n22. Yes\n23. No (see proof)\n24. Yes\n25. No (see proof)\n26. Yes\n27. Yes\n28. Yes\n29.\n30.\n31.\n     (a) No\n     (b) No\n     (c) No\n     (d) No\n32.",
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      "title": "[Apostol's Calculus] Exercises 1.5",
      "body": "@@ -681,24 +681,25 @@\n  Yes%0A5. No (\n+%5B\n see proof)%0A6\n@@ -695,16 +695,106 @@\n ee proof\n+%5D(https://steemit.com/mathematics/@shinday/apostol-s-calculus-exercises-1-5-solution-to-5)\n )%0A6. Yes\n",
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      "title": "[Apostol's Calculus] Exercises 1.5: Solution to 5",
      "body": "Let S be the set of all functions _f_ such that _f_(1) = 1 + _f_(0). Assume that S is a real linear space. Then S must be closed under addition. In other words, if g(x) is in S, then the following must hold:\n\ng(x) ∈ S \n(g(x) + g(x)) ∈ S\n2g(x) ∈ S\n\nNow let's define the following:\n\n(2g(x) = h(x))\n\nNotice that since we're assuming that S is a real linear space, then h(x) must be in S, since real linear spaces are closed under addition. Furthermore, we were told that all functions _f_ in S must satisfy the property _f_(1) = 1 + _f_(0). Thus,\n\nh(1) = 1 + h(0)\n\nBut h(x) = 2g(x). So,\n\n2g(1) = 1 + 2g(0)\ng(1) = 0.5 + g(0)\n\nBut this violates the condition of being in S, and we assumed that g(x) was in S. So S cannot possibly be closed under addition.\n\nIt follows that S cannot possibly be a real linear space, since it is not closed under addition.\n\nQED",
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      "author": "shinday",
      "permlink": "apostol-s-calculus-exercises-1-5",
      "title": "[Apostol's Calculus] Exercises 1.5",
      "body": "The key to solving these exercises is to analyze each of the given sets and determine whether or not they violate any of the 10 axioms of real linear vectors spaces V<sub>n</sub>:\n1. Closure under addition\n2. Closure under multiplication\n3. Commutativity of addition\n4. Associativity of Addition\n5. Existence of 0 element\n6. Existence of negatives\n7. associativity of scalar multiplication\n8. Distributive law for addition in V\n9. Distributive law for addition of numbers\n10. Existence of Identity Element\n\nFor these solutions, I will simply state \"Yes\" if the example given is a real linear space, but I will provide a formal proof if it is not.\n\nAnswers:\n1. Yes\n2. Yes\n3. Yes\n4. Yes\n5. No (see proof)\n6. Yes\n7. Yes\n8. Yes\n9. Yes\n10. Yes\n11. No (see proof)\n12. Yes\n13. Yes\n14. No (see proof)\n15. Yes\n16. Yes\n17. Yes\n18. Yes\n19. Yes\n20. Yes\n21. Yes\n22. Yes\n23. No (see proof)\n24. Yes\n25. No (see proof)\n26. Yes\n27. Yes\n28. Yes\n29.\n30.\n31.\n     (a) No\n     (b) No\n     (c) No\n     (d) No\n32.",
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      "body": "**Please note this is a work in progress. I will update this page (with links) as I gather more solutions. This is an ongoing project that I expect will take close to a year to finish.**\n\n# Links:\n##### Chapter 1\n- 1.5\n- 1.10 \n- 1.13\n- 1.17\n##### Chapter 2\n- 2.4\n- 2.8\n- 2.12\n- 2.16\n- 2.20\n- 2.21\n##### Chapter 3\n- 3.6\n- 3.11\n- 3.17\n##### Chapter 4\n- 4.4\n- 4.8\n- 4.10",
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