What are vector spaces?
Let (F,+,.) be a field. The element of F will be called scalars. Let V be a non-empty set whose elements will be called a vector. Then V is a vector space over the field F.
If an internal composition in V called the addition of vectors and denoted by'+'.Also for this composition Vis an abelian group.
An external composition in V over F called scalar multiplication and denoted by multiplicatively so that ax⋿V for all a⋿ F and for all 𝞪⋿V(V is close to scalar multiplication)
Vectors apply in the programming language R
# creat the vector
> x<-1:30
> x
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
[16] 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
> y<-29:1
> y
[1] 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
[16] 14 13 12 11 10 9 8 7 6 5 4 3 2 1
> z<-(1:30,29:1)
Error: unexpected ',' in "z<-(1:30,"
> z<-c(1:30,29:1)
> z
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
[16] 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
[31] 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
[46] 14 13 12 11 10 9 8 7 6 5 4 3 2 1
> tep<-c(4,6,3)
> tep
[1] 4 6 3
> 3^tep
[1] 81 729 27
> 7^tep
[1] 2401 117649 343
> tep^(2+tep)
[1] 4096 1679616 243
> tep^(2+tep^3)
[1] 5.444518e+39 4.334835e+169 6.863038e+13
> rep(tep,10)
[1] 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6
[24] 3 4 6 3 4 6 3
> rep(tep,l=31)
[1] 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6 3 4 6
[24] 3 4 6 3 4 6 3 4
> #create a vector of the values of e^x cos(x)at x=4,4.1,4.2....7
> qap<-seq(4,7,by=0.1)
> exp(qap)*cos(qap)
[1] -35.687732 -34.685042 -32.693695 -29.538816
[5] -25.032529 -18.975233 -11.157417 -1.362099
[9] 10.632038 25.046705 42.099201 61.996630
[13] 84.929067 111.061586 140.525075 173.405776
[17] 209.733494 249.468441 292.486707 338.564378
[21] 387.360340 438.397873 491.045163 544.494927
[25] 597.743415 649.569088 698.511407 742.850196
[29] 780.586182 809.423392 826.754210
> exp(qap)*sin(qap)
[1] -41.320016 -49.375076 -58.122190
[4] -67.521241 -77.508816 -87.994457
[7] -98.856669 -109.938735 -121.044378
[10] -131.933345 -142.316981 -151.853890
[13] -160.145806 -166.733804 -171.095016
[16] -172.640026 -170.711169 -164.581957
[19] -153.457895 -136.478991 -112.724257
[22] -81.218568 -40.942224 9.156378
[25] 70.144561 143.085260 229.012558
[28] 328.902434 443.638334 573.971153
[31] 720.473289
> tanh(qap)
[1] 0.9993293 0.9994508 0.9995504 0.9996319
[5] 0.9996986 0.9997532 0.9997979 0.9998346
[9] 0.9998646 0.9998891 0.9999092 0.9999257
[13] 0.9999391 0.9999502 0.9999592 0.9999666
[17] 0.9999727 0.9999776 0.9999817 0.9999850
[21] 0.9999877 0.9999899 0.9999918 0.9999933
[25] 0.9999945 0.9999955 0.9999963 0.9999970
[29] 0.9999975 0.9999980 0.9999983
> exp(tanh(qap))
[1] 2.716459 2.716789 2.717060 2.717281 2.717463
[6] 2.717611 2.717733 2.717832 2.717914 2.717980
[11] 2.718035 2.718080 2.718116 2.718146 2.718171
[16] 2.718191 2.718207 2.718221 2.718232 2.718241
[21] 2.718248 2.718254 2.718259 2.718263 2.718267
[26] 2.718270 2.718272 2.718274 2.718275 2.718276
[31] 2.718277
> #creat vectors
> #(0.1^3 0.2^1,0.1^6 0.2^4.....,0.1^36 0.2^34 )
> (0.1^seq(3,36,by=3))*(0.2^seq(1,34,by=3))
[1] 2.000000e-04 1.600000e-09 1.280000e-14
[4] 1.024000e-19 8.192000e-25 6.553600e-30
[7] 5.242880e-35 4.194304e-40 3.355443e-45
[10] 2.684355e-50 2.147484e-55 1.717987e-60
> #(2,2^2/2,2^3/3,......,2^25/25)
> (2^(1:25))/(1:25)
[1] 2.000000e+00 2.000000e+00 2.666667e+00
[4] 4.000000e+00 6.400000e+00 1.066667e+01
[7] 1.828571e+01 3.200000e+01 5.688889e+01
[10] 1.024000e+02 1.861818e+02 3.413333e+02
[13] 6.301538e+02 1.170286e+03 2.184533e+03
[16] 4.096000e+03 7.710118e+03 1.456356e+04
[19] 2.759411e+04 5.242880e+04 9.986438e+04
[22] 1.906502e+05 3.647221e+05 6.990507e+05
[25] 1.342177e+06
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