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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
How do you show the direct sum of the image and kernel of the linear operator is the vector space?
Ok, linear algebra isn't my strongest area but I'll have a go (please note that all vectors are in bold, if a mathematical lower case letter is not in bold, assume it to be a scalar unless otherwise indicated).
First, I'm going to assume that you already know how to prove that the kernel and image of a linear operator, say f:U->V where U and V are vector spaces, is a vector subspace of U as this needs to be proved before going on to show that the sum of ker(f) and Im(f) are a vector space.
Now, a vector space must follow the following axioms (here A represents the addition axioms, and M the multiplication axioms):
A1) associativity u+(v+w)=(u+v)+w
A2) commutativity u+v=v+u
A3) identity there exists an element 0 in U such that v+0=v
A4) existence of an inverse for all u in U there exists an element -u in U such that u+(-u)=0
M1) scalar multiplication with respect to vector addition a(u+v)=au+av
M2) scalar multiplication with respect to field addition (a+b)u=au+bu
M3) compatibility of scalar and field multiplication (ab)u=a(bu)
M4) identity 1u=u where 1 is the multiplicative identity
ker(f)={u in U | f(u)=0} (or more strictly, the set of vectors u in U such that Au=0 i.e. ker(f) is the null space of the matrix A), Im(f)={v in V | f(u)=v for some u in U}
Let ker(f)+Im(f) = W. Show W is a vector space i.e. show W satisfies the above axioms.
A1, A2, M1, M2, and M3 are all trivial/simple to prove.
A3: since Im(f) is the set off all v in V obtained from f(u), assume Im(f) = V and as V is a vector space it must have, by definition, an additive identity 0, therefore W also contains 0.
A4: since ker(f) is the null space of the matrix A, f(0)=0. Now assume that f(u)=0 and f(-u)=v, since U is a vector space u+(-u)=0 so f(u+(-u))=f(0)=0 but f(u+(-u))=f(u)+f(-u)=0+vtherefore, 0=0+v which implies v=0, so if u goes to 0 in V, its inverse -u also goes to 0. A similar process for the image shows that if u goes to v then -u goes to -v. (Thanks to Jokes Free4Me for the help with this axiom)
M4: similar to A3 except V, again by definition, must have the multiplicative identity 1 and so also exists in W.
All the axioms are satisfied, therefore W=ker(f)+Im(f) is a vector space. Q.E.D.
Abe sale dimag kharab mat kar samjha>>>>>>>>>>>>>>>>
Let z be a complex number such that Imz 0 Prove that Im1z 0?
If the imaginary part of z is 0, then z is simply a real number and if you multiply by 1 which is the identity in multiplication of real numbers, you of course will still have a real number with imaginary part 0
A parallelogram has symmetry with respect to the point of intersection of its diagonals?
Yes, rotational symmetry of order 2.
Prime numbers not a factor of 12?
2 and 3 are prime numbers that are factors of 12. There is an infinite list of prime numbers that aren't.
They share a common base, CD, and their height is the distance between the parallel lines AB and CD.
Consequently, area(ACD) = area(BCD) . . . . . . . . eqn 1
But ACD = AOD + OCD and BCD = BOC + OCD
So area(ACD) = area(AOD) + area(OCD)
and area(BCD) = area(BOC) + area(OCD)
Substituting in eqn 1,
area(AOD) + area(OCD) = area(BOC) + area(OCD)
area(AOD) = area(BOC)
What is the next number in a sequence 60 32 18?
This is a highly ambiguous question, because there are many ways to continue any finite sequence. In fact, there are infinitely many, because any number whatsoever can be the next number in some sequence. In this case, I can think of at least 2 that make sense.
- 32 is 28 less than 60, and 18 is 14 short of 32. So first they subtract 28, then 14, which is half of 28. If u continue that logic, u would take half of 14 (7) and subtract it from 18, so the next number would be 11.
- Again, they subtract 28 and then 14, which 14 less than 28. So the next number to be subtracted should be 14 - 14 = 0. So the next number should be 18 again.
I could go on and on, but u get the idea.
2 and 1/2 are rational numbers, but 2^(1/2) is the square root of 2. It is well known that the square root of 2 is not rational.
What does pythagarean theorem state?
In a right triangle the square of hypotenuse is equal to the sum of squares of the other two sides
Prove by mathematical induction that 3n-1 is divisible by 2?
It's not. If n = 2, then 3n - 1 = 3*2 - 1 = 6 - 1 = 5, which isn't divisible by 2.
How many gallons do you pee yearly?
Somewhere between 80 and 220 gallons of urine yearly (.8 to 2.2 liters daily)
100 million
How do you prove the HL postulate?
My geometry teacher uploads his lessons to YouTube.
The proof itself starts at 1:28.
Find the product of two numbers Now reverse the digits and find the product being the same?
The trivial answer is that any two numbers that reverse to each other will have this property. For example:
19 * 91 = 91 * 19
237 * 732 = 732 * 237
etc.
But apart from those, here are the two-digit pairs where this works:
12*42 = 21*24 = 504
12*63 = 21*36 = 756
12*84 = 21*48 = 1008
13*62 = 31*26 = 806
13*93 = 31*39 = 1209
14*82 = 41*28 = 1148
21*24 = 12*42 = 504
21*36 = 12*63 = 756
21*48 = 12*84 = 1008
23*64 = 32*46 = 1472
23*96 = 32*69 = 2208
24*63 = 42*36 = 1512
24*84 = 42*48 = 2016
26*31 = 62*13 = 806
26*93 = 62*39 = 2418
28*41 = 82*14 = 1148
31*39 = 13*93 = 1209
32*46 = 23*64 = 1472
32*69 = 23*96 = 2208
34*86 = 43*68 = 2924
36*42 = 63*24 = 1512
36*84 = 63*48 = 3024
39*62 = 93*26 = 2418
42*48 = 24*84 = 2016
43*68 = 34*86 = 2924
46*96 = 64*69 = 4416
48*63 = 84*36 = 3024
64*69 = 46*96 = 4416
Here are the two-digit numbers paired with three-digit numbers where this works:
12*231 = 21*132 = 2772
12*462 = 21*264 = 5544
12*693 = 21*396 = 8316
13*341 = 31*143 = 4433
13*682 = 31*286 = 8866
14*451 = 41*154 = 6314
15*561 = 51*165 = 8415
16*671 = 61*176 = 10736
17*781 = 71*187 = 13277
18*891 = 81*198 = 16038
21*132 = 12*231 = 2772
21*264 = 12*462 = 5544
21*396 = 12*693 = 8316
23*352 = 32*253 = 8096
24*231 = 42*132 = 5544
24*462 = 42*264 = 11088
24*693 = 42*396 = 16632
25*572 = 52*275 = 14300
26*341 = 62*143 = 8866
26*682 = 62*286 = 17732
27*792 = 72*297 = 21384
28*451 = 82*154 = 12628
31*143 = 13*341 = 4433
31*286 = 13*682 = 8866
32*253 = 23*352 = 8096
34*473 = 43*374 = 16082
35*583 = 53*385 = 20405
36*231 = 63*132 = 8316
36*462 = 63*264 = 16632
36*693 = 63*396 = 24948
39*341 = 93*143 = 13299
39*682 = 93*286 = 26598
41*154 = 14*451 = 6314
42*132 = 24*231 = 5544
42*264 = 24*462 = 11088
42*396 = 24*693 = 16632
43*374 = 34*473 = 16082
45*594 = 54*495 = 26730
46*352 = 64*253 = 16192
48*231 = 84*132 = 11088
48*462 = 84*264 = 22176
48*693 = 84*396 = 33264
51*165 = 15*561 = 8415
52*275 = 25*572 = 14300
53*385 = 35*583 = 20405
54*495 = 45*594 = 26730
61*176 = 16*671 = 10736
62*143 = 26*341 = 8866
62*286 = 26*682 = 17732
63*132 = 36*231 = 8316
63*264 = 36*462 = 16632
63*396 = 36*693 = 24948
64*253 = 46*352 = 16192
68*473 = 86*374 = 32164
69*352 = 96*253 = 24288
71*187 = 17*781 = 13277
72*297 = 27*792 = 21384
81*198 = 18*891 = 16038
82*154 = 28*451 = 12628
84*132 = 48*231 = 11088
84*264 = 48*462 = 22176
84*396 = 48*693 = 33264
86*374 = 68*473 = 32164
93*143 = 39*341 = 13299
93*286 = 39*682 = 26598
96*253 = 69*352 = 24288
Here are the three-digit numbers where this works:
102*402 = 201*204 = 41004
102*603 = 201*306 = 61506
102*804 = 201*408 = 82008
103*602 = 301*206 = 62006
103*903 = 301*309 = 93009
104*802 = 401*208 = 83408
112*422 = 211*224 = 47264
112*633 = 211*336 = 70896
112*844 = 211*448 = 94528
113*622 = 311*226 = 70286
113*933 = 311*339 = 105429
114*822 = 411*228 = 93708
122*442 = 221*244 = 53924
122*663 = 221*366 = 80886
122*884 = 221*488 = 107848
123*642 = 321*246 = 78966
123*963 = 321*369 = 118449
124*842 = 421*248 = 104408
132*462 = 231*264 = 60984
132*693 = 231*396 = 91476
133*662 = 331*266 = 88046
133*993 = 331*399 = 132069
134*862 = 431*268 = 115508
142*482 = 241*284 = 68444
143*682 = 341*286 = 97526
144*882 = 441*288 = 127008
201*204 = 102*402 = 41004
201*306 = 102*603 = 61506
201*408 = 102*804 = 82008
203*604 = 302*406 = 122612
203*906 = 302*609 = 183918
204*603 = 402*306 = 123012
204*804 = 402*408 = 164016
206*301 = 602*103 = 62006
206*903 = 602*309 = 186018
208*401 = 802*104 = 83408
211*224 = 112*422 = 47264
211*336 = 112*633 = 70896
211*448 = 112*844 = 94528
213*624 = 312*426 = 132912
213*936 = 312*639 = 199368
214*824 = 412*428 = 176336
221*244 = 122*442 = 53924
221*366 = 122*663 = 80886
221*488 = 122*884 = 107848
223*644 = 322*446 = 143612
223*966 = 322*669 = 215418
224*633 = 422*336 = 141792
224*844 = 422*448 = 189056
226*311 = 622*113 = 70286
226*933 = 622*339 = 210858
228*411 = 822*114 = 93708
231*264 = 132*462 = 60984
231*396 = 132*693 = 91476
233*664 = 332*466 = 154712
233*996 = 332*699 = 232068
234*864 = 432*468 = 202176
241*284 = 142*482 = 68444
243*684 = 342*486 = 166212
244*663 = 442*366 = 161772
244*884 = 442*488 = 215696
246*321 = 642*123 = 78966
246*963 = 642*369 = 236898
248*421 = 842*124 = 104408
264*693 = 462*396 = 182952
266*331 = 662*133 = 88046
266*993 = 662*399 = 264138
268*431 = 862*134 = 115508
286*341 = 682*143 = 97526
288*441 = 882*144 = 127008
301*309 = 103*903 = 93009
302*406 = 203*604 = 122612
302*609 = 203*906 = 183918
304*806 = 403*608 = 245024
306*402 = 603*204 = 123012
306*804 = 603*408 = 246024
309*602 = 903*206 = 186018
311*339 = 113*933 = 105429
312*426 = 213*624 = 132912
312*639 = 213*936 = 199368
314*826 = 413*628 = 259364
321*369 = 123*963 = 118449
322*446 = 223*644 = 143612
322*669 = 223*966 = 215418
324*846 = 423*648 = 274104
331*399 = 133*993 = 132069
332*466 = 233*664 = 154712
332*699 = 233*996 = 232068
334*866 = 433*668 = 289244
336*422 = 633*224 = 141792
336*844 = 633*448 = 283584
339*622 = 933*226 = 210858
342*486 = 243*684 = 166212
344*886 = 443*688 = 304784
366*442 = 663*244 = 161772
366*884 = 663*488 = 323544
369*642 = 963*246 = 236898
396*462 = 693*264 = 182952
399*662 = 993*266 = 264138
402*408 = 204*804 = 164016
403*608 = 304*806 = 245024
406*906 = 604*609 = 367836
408*603 = 804*306 = 246024
412*428 = 214*824 = 176336
413*628 = 314*826 = 259364
422*448 = 224*844 = 189056
423*648 = 324*846 = 274104
426*936 = 624*639 = 398736
432*468 = 234*864 = 202176
433*668 = 334*866 = 289244
442*488 = 244*884 = 215696
443*688 = 344*886 = 304784
446*966 = 644*669 = 430836
448*633 = 844*336 = 283584
466*996 = 664*699 = 464136
488*663 = 884*366 = 323544
604*609 = 406*906 = 367836
624*639 = 426*936 = 398736
644*669 = 446*966 = 430836
664*699 = 466*996 = 464136
The relationship between triangles and circles?
The three vertices of the triangle uniquely determine a circle that circumscribes the triangle. The three sides of the triangle uniquely determine the circle that inscribes the triangle.
Is this equation solvable 3x plus 4y-6c equals -7 and x plus 9y equals 19 and 5x-y equals 3?
YES : Equation (1) 5x - y = 3
x + 9y = 19 : Multiply by 5 →Equation (2) 5x + 45y = 95
Subtract Equation (1) from Equation (2) to eliminate 'x' terms. This gives : 45y -(-y) = 95 - 3 : 46y = 92 : y = 2
Substituting for y in Equation(1) gives : 5x - 2 = 3 : 5x = 3 + 2 = 5 : x = 1
Substituting for x and y in the equation 3x + 4y - 6c = -7 gives : 3 + 8 - 6c = -7 : 6c = 11 + 7 = 18 : c = 3
Addition and subtraction of radicals?
solving problems on exponents and radicals have come seriously to my life didn't knew anything but when solved algebraic way like it was marvellous should just use litte brain and god for sure dont cry if you dont know.Here it goes
Like,
2a+3a=5a
so, 3root 5 + 4 root 5=7 root 5
sorry since i can't find the sign of root, i wrote
