05.Matrices.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = reshape(1:12, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 Array{Float64,2}:\n",
       " 1.0  1.0  1.0  1.0\n",
       " 1.0  1.0  1.0  1.0\n",
       " 1.0  1.0  1.0  1.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = ones(3,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 Array{Float64,2}:\n",
       " 2.0  5.0   8.0  11.0\n",
       " 3.0  6.0   9.0  12.0\n",
       " 4.0  7.0  10.0  13.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A+B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 Array{Float64,2}:\n",
       " 0.0  3.0  6.0   9.0\n",
       " 1.0  4.0  7.0  10.0\n",
       " 2.0  5.0  8.0  11.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A-B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A==B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching isless(::Array{Float64,2}, ::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}})\nClosest candidates are:\n  isless(!Matched::Missing, ::Any) at missing.jl:66\n  isless(::Any, !Matched::Missing) at missing.jl:67",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching isless(::Array{Float64,2}, ::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}})\nClosest candidates are:\n  isless(!Matched::Missing, ::Any) at missing.jl:66\n  isless(::Any, !Matched::Missing) at missing.jl:67",
      "",
      "Stacktrace:",
      " [1] <(::Array{Float64,2}, ::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}) at ./operators.jl:260",
      " [2] >(::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}, ::Array{Float64,2}) at ./operators.jl:286",
      " [3] top-level scope at In[7]:1"
     ]
    }
   ],
   "source": [
    "A>B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " 1.83333  1.83333  1.83333\n",
       " 2.16667  2.16667  2.16667\n",
       " 2.5      2.5      2.5    "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A/B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "ename": "DimensionMismatch",
     "evalue": "DimensionMismatch(\"matrix A has dimensions (3,4), matrix B has dimensions (3,4)\")",
     "output_type": "error",
     "traceback": [
      "DimensionMismatch(\"matrix A has dimensions (3,4), matrix B has dimensions (3,4)\")",
      "",
      "Stacktrace:",
      " [1] _generic_matmatmul!(::Array{Float64,2}, ::Char, ::Char, ::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}, ::Array{Float64,2}) at /build/julia-WtZlcm/julia-1.0.4+dfsg/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/matmul.jl:591",
      " [2] generic_matmatmul! at /build/julia-WtZlcm/julia-1.0.4+dfsg/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/matmul.jl:581 [inlined]",
      " [3] mul! at /build/julia-WtZlcm/julia-1.0.4+dfsg/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/matmul.jl:175 [inlined]",
      " [4] *(::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}, ::Array{Float64,2}) at /build/julia-WtZlcm/julia-1.0.4+dfsg/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/matmul.jl:142",
      " [5] top-level scope at In[9]:1"
     ]
    }
   ],
   "source": [
    "A*B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching rem(::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}, ::Array{Float64,2})\nClosest candidates are:\n  rem(::Any, ::Any, !Matched::RoundingMode{:ToZero}) at math.jl:739\n  rem(::Any, ::Any, !Matched::RoundingMode{:Down}) at math.jl:740\n  rem(::Any, ::Any, !Matched::RoundingMode{:Up}) at math.jl:741\n  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching rem(::Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}, ::Array{Float64,2})\nClosest candidates are:\n  rem(::Any, ::Any, !Matched::RoundingMode{:ToZero}) at math.jl:739\n  rem(::Any, ::Any, !Matched::RoundingMode{:Down}) at math.jl:740\n  rem(::Any, ::Any, !Matched::RoundingMode{:Up}) at math.jl:741\n  ...",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[10]:1"
     ]
    }
   ],
   "source": [
    "A%B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 1  2\n",
       " 3  4"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1 2 ; 3 4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 10  11\n",
       " 12  13"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = [10 11 ; 12 13]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 10  22\n",
       " 36  52"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A .* B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 34  37\n",
       " 78  85"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 34  37\n",
       " 78  85"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1 * 10 + 2 * 12        1 * 11 + 2 * 13  ;      3 * 10 + 4 * 12     3 * 11 + 4 * 13]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       " 4  3  1\n",
       " 7  3  6\n",
       " 1  4  9"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(1:9, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       " 1  6  9\n",
       " 7  2  7\n",
       " 1  1  6"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = rand(1:9, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       "  1.06716    0.0970149   0.328358\n",
       " -1.27612    2.15672     2.76119 \n",
       "  0.559701  -0.858209   -0.597015"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A \\ B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " 0.5      0.7     -1.4\n",
       " 0.25     1.05    -0.6\n",
       " 0.5625  -0.0375   0.7"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A/B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching +(::Array{Int64,2}, ::Int64)\nClosest candidates are:\n  +(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:502\n  +(!Matched::Complex{Bool}, ::Real) at complex.jl:292\n  +(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching +(::Array{Int64,2}, ::Int64)\nClosest candidates are:\n  +(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:502\n  +(!Matched::Complex{Bool}, ::Real) at complex.jl:292\n  +(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[20]:1"
     ]
    }
   ],
   "source": [
    "A + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×5 Array{Int64,2}:\n",
       " 2  4  6  8  10"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1 2 3 4 5] * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       " 3  2  0\n",
       " 6  2  5\n",
       " 0  3  8"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A .- 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       "  8  6   2\n",
       " 14  6  12\n",
       "  2  8  18"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A .* 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " 2.0  1.5  0.5\n",
       " 3.5  1.5  3.0\n",
       " 0.5  2.0  4.5"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A ./ 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Rational{Int64},2}:\n",
       " 2//1  3//2  1//2\n",
       " 7//2  3//2  3//1\n",
       " 1//2  2//1  9//2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A // 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 BitArray{2}:\n",
       "  true  true   true\n",
       " false  true  false\n",
       "  true  true  false"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A .< 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = reshape(1:12, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 2\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = collect(1:4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Int64,1}:\n",
       " 70\n",
       " 80\n",
       " 90"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A*V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "134"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot([1:3...], [21:23...])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "134"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1 * 21) + (2 * 22) +  (3 * 23)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "134"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1:3] ⋅ [21:23]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = reshape(1:12, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::StepRange{Int64,Int64}, 3, 4) with eltype Int64:\n",
       " 100  400  700  1000\n",
       " 200  500  800  1100\n",
       " 300  600  900  1200"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = reshape(100:100:1200, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×8 Array{Int64,2}:\n",
       " 1  4  7  10  100  400  700  1000\n",
       " 2  5  8  11  200  500  800  1100\n",
       " 3  6  9  12  300  600  900  1200"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hcat(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×4 Array{Int64,2}:\n",
       "   1    4    7    10\n",
       "   2    5    8    11\n",
       "   3    6    9    12\n",
       " 100  400  700  1000\n",
       " 200  500  800  1100\n",
       " 300  600  900  1200"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vcat(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12-element Array{Float64,1}:\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vec(ones(3, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Array,1}:\n",
       " [1, 2]\n",
       " [3, 4]\n",
       " [5, 6]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Array[[1, 2], [3, 4], [5, 6]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×3 Array{Int64,2}:\n",
       " 1  3  5\n",
       " 2  4  6"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hcat(a...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0-element Array{Any,1}"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i = 1:80\n",
    "    push!(a, i)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "80-element Array{Any,1}:\n",
       "  1\n",
       "  2\n",
       "  3\n",
       "  4\n",
       "  5\n",
       "  6\n",
       "  7\n",
       "  8\n",
       "  9\n",
       " 10\n",
       " 11\n",
       " 12\n",
       " 13\n",
       "  ⋮\n",
       " 69\n",
       " 70\n",
       " 71\n",
       " 72\n",
       " 73\n",
       " 74\n",
       " 75\n",
       " 76\n",
       " 77\n",
       " 78\n",
       " 79\n",
       " 80"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10×8 Array{Any,2}:\n",
       "  1  11  21  31  41  51  61  71\n",
       "  2  12  22  32  42  52  62  72\n",
       "  3  13  23  33  43  53  63  73\n",
       "  4  14  24  34  44  54  64  74\n",
       "  5  15  25  35  45  55  65  75\n",
       "  6  16  26  36  46  56  66  76\n",
       "  7  17  27  37  47  57  67  77\n",
       "  8  18  28  38  48  58  68  78\n",
       "  9  19  29  39  49  59  69  79\n",
       " 10  20  30  40  50  60  70  80"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reshape(a, 10, :)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 reshape(::UnitRange{Int64}, 3, 4) with eltype Int64:\n",
       " 1  4  7  10\n",
       " 2  5  8  11\n",
       " 3  6  9  12"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = reshape(1:12, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×3 Transpose{Int64,Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}}:\n",
       "  1   2   3\n",
       "  4   5   6\n",
       "  7   8   9\n",
       " 10  11  12"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transpose(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×3 Adjoint{Int64,Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}}:\n",
       "  1   2   3\n",
       "  4   5   6\n",
       "  7   8   9\n",
       " 10  11  12"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M' "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0000000000000284"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = [10 11 ; 12 13]\n",
    "det(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " -6.5   5.5\n",
       "  6.0  -5.0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inv(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rank(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 10  11\n",
       " 12  13"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×0 Array{Float64,2}"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nullspace(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 reshape(::UnitRange{Int64}, 3, 3) with eltype Int64:\n",
       " 1  4  7\n",
       " 2  5  8\n",
       " 3  6  9"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = reshape(1:9, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tr(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 reshape(::UnitRange{Int64}, 3, 3) with eltype Int64:\n",
       " 1  4  7\n",
       " 2  5  8\n",
       " 3  6  9"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = reshape(1:9, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×3 Array{Int64,2}:\n",
       " 6  15  24"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(A, dims=(1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×1 Array{Int64,2}:\n",
       " 12\n",
       " 15\n",
       " 18"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(A, dims=(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.0.4",
   "language": "julia",
   "name": "julia-1.0"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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