Day 1 almost done

Day_1/Day_1.jl

@@ -4,6 +4,11 @@
 using Markdown
 using InteractiveUtils
 
+# ╔═╡ 1f347724-1db2-48f0-87df-4e63ad6e8820
+# Importing a builtin library that provides more functions for linear algebra.
+# The keyword `using` imports the package and exports (public) functions automatically.
+using LinearAlgebra
+
 # ╔═╡ d1a4ef8b-8e7d-4d34-80d8-cee195e237ae
 begin
 	using PlutoUI
@@ -13,7 +18,7 @@ end
 
 # ╔═╡ 2c5e32f4-1d7d-4494-b025-a90d17919756
 md"""
-# Introduction to Julia
+# Introduction
 """
 
 # ╔═╡ 21590bf1-1e1c-46b4-a2b6-7eb915e121ab
@@ -43,6 +48,11 @@ md"""
 - 📚️ Take a look at the **live docs** to the right.
 """
 
+# ╔═╡ b559ef1a-76cf-4464-b5ec-f7d6bfa892e6
+md"""
+# Basics
+"""
+
 # ╔═╡ 938adcfe-8d1b-4c77-8d82-c48415f5673e
 md"""
 ## Calculation
@@ -164,6 +174,62 @@ meaning_of_life = 42
 # Int64 is the default
 typeof(meaning_of_life)
 
+# ╔═╡ 944e2d37-8280-47b8-b874-97221955d048
+md"""
+## Type hierarchy
+
+Julia does have abstract types which are helpful for multiple dispatch.
+
+More about multiple dispatch later 😉
+"""
+
+# ╔═╡ 23d4ac67-05ec-4b3d-8368-86256076be62
+# The type hierarchy of Int64
+supertypes(Int64)
+
+# ╔═╡ e8d7de2f-7c7e-47cb-9364-27d583652167
+md"""
+All types showed in the output of the above cell except `Int64` are abstract.
+
+This means that you can not have a variable with an abstract type.
+
+You can only derive from an abstract type, but more about this when discussing structs and mutliple dispatch 😉
+
+`Any` is the abstract type of everything.
+"""
+
+# ╔═╡ 866edf5e-a76c-448d-98e8-925eaed5eba5
+# A number can either be real or complex
+subtypes(Number)
+
+# ╔═╡ a8ea4ac1-7f62-4485-a8c5-8ccf00c45720
+# There are some types of real numbers.
+subtypes(Real)
+
+# ╔═╡ 786a96be-16cd-4f1b-9b5f-138e232d3183
+# Integers can have a sign or no sign (only positive).
+# A Bool is also treated as an integer with the value 0 or 1.
+subtypes(Integer)
+
+# ╔═╡ dd8dad86-6bc2-4489-8469-7eac80fc41bb
+# Integers can have different number of bits.
+subtypes(Signed)
+
+# ╔═╡ fac04aa7-28e9-4f93-9312-a8f8f93c0877
+# Minimum and maximum value of a type
+typemin(Int8), typemax(Int8)
+
+# ╔═╡ 86bc0ff0-b6bf-4700-a741-36323be58391
+typemin(Int128), typemax(Int128)
+
+# ╔═╡ ef35a3a7-c1df-4952-aa41-1ed22d7f3981
+# BigInt does not have a minimum or maximum!
+BigInt(10)^1000 + 1
+
+# ╔═╡ c336d5f6-80ee-4994-a55f-2d6b3aa3d559
+# Hierarchy of Float64
+supertypes(Float64)
+
 # ╔═╡ 608d4433-6e68-4f95-8581-437234b58e87
 md"""
 ## Convertion
@@ -288,22 +354,110 @@ a_mult_b = a * b
 # Using the macro @show, helpful for usage in scripts
 @show a * b
 
+# ╔═╡ 3102810f-3467-4ed8-86c0-16e9177fa69d
+md"""
+## For loop
+"""
+
+# ╔═╡ c1c705d2-7e46-4811-9fb1-6b88b5a4140e
+for i in 1:3
+	println(i)
+end
+
+# ╔═╡ 00456b15-5d1c-4c74-a875-31ff9c8e1789
+md"""
+## While loop
+"""
+
+# ╔═╡ 91c4c623-5680-4b35-a694-2bd2612def94
+begin
+	value = 5
+	while value > 0
+		value -= 1
+		println(value)
+	end
+end
+
+# ╔═╡ 2a85d95b-51d2-4ea0-a2a2-43307a725f2a
+md"""
+## If, else, elseif
+"""
+
+# ╔═╡ 9a78bf14-7fb4-448a-a8dd-69e244a0a297
+test_value = 1
+
+# ╔═╡ 5281ef32-5de6-4488-8430-e5652cbf8299
+if test_value == 1
+	println("The value is 1")
+end
+
+# ╔═╡ eec41279-e038-4415-81b3-ad5d4c396011
+# change the value of the variable `test_value` and see how the input changes
+if test_value == 1
+	println("The value is 1")
+else
+	println("The value is not 1")
+end
+
+# ╔═╡ d2333817-e941-429b-b8e3-2ff07669096b
+# change the value of the variable `test_value` and see how the input changes
+if test_value == 1
+	println("The value is 1")
+elseif test_value == 2
+	println("The value is 2")
+elseif test_value == 3
+	println("The value is 3")
+else
+	println("The value is not 1, 2 or 3")
+end
+
+# ╔═╡ be0ff87b-229a-433e-a49e-2f1ced5bb9aa
+# You can combine conditions
+if (test_value == 1) || (test_value == 2)
+	println("Value is 1 or 2")
+end
+
+# ╔═╡ 96803a1e-0779-4eab-b120-b5569a44ac7b
+second_test_value = 2
+
+# ╔═╡ 5ab233d2-f360-4362-b1f8-3f3ae2a4fee1
+# change the value of the variable `second_test_value` and see how the input changes
+if (test_value == 1) && (second_test_value == 1)
+	println("Both values are 1")
+end
+
+# ╔═╡ 0d100501-de84-4a5c-beb7-8ff9e83c473d
+# change the value of the variable `test_value` and see how the input changes
+if !(test_value == 1)
+	println("Value is not 1")
+end
+
 # ╔═╡ 9e3f698d-e57b-46c2-98e0-157fa7b06ae6
 md"""
-## Arrays
+# Arrays
 An array is a **mutable ordered** collection of elements of the same type.
 
-Arrays can have one dimension (vector), two dimensions (matrix) or more (tensor).
+Arrays can have different dimensions.
+
+An array with (`n × 1`) dimensions is called a vector, like in mathematics.
+
+An array with (`n × n`) dimensions is called a matrix, also like in mathematics.
+
+But arrays can also have other dimensions (`n₁ × n₂ × n₃ × ...`) with `nᵢ` as natural numbers.
 """
 
 # ╔═╡ d1680205-a8eb-4ef6-ae4f-059e7a30f5c1
 md"""
-### Vectors
+## Vectors
 """
 
 # ╔═╡ b54ace0e-8947-46a3-842a-05b5cbfc4e87
 first_vector = [2, 4, 6, 8, 10]
 
+# ╔═╡ 0d5bfd45-79da-435b-9a98-8ed996bbc7b4
+# Show the dimensions of an array
+size(first_vector)
+
 # ╔═╡ 8c47710f-1ed2-40fd-9290-374b498380e3
 first_vector[1]
 
@@ -332,7 +486,7 @@ first_vector[end-1]
 
 # ╔═╡ 6c80e009-30de-4232-9a1b-ac954242a5a6
 md"""
-### Slicing
+## Slicing
 """
 
 # ╔═╡ b1426df5-a083-4977-a72c-81e03fd7719d
@@ -349,7 +503,7 @@ first_vector[[1, end, 2]]
 
 # ╔═╡ 3ea54f0d-2aa5-47a3-bbc3-92023a56b834
 md"""
-### Mutation
+## Mutation
 """
 
 # ╔═╡ 6fdc6add-a478-4707-876b-cf6d660870ba
@@ -390,33 +544,6 @@ end
 # But then, the type of the array is Any
 typeof(mixed_vector)
 
-# ╔═╡ 944e2d37-8280-47b8-b874-97221955d048
-md"""
-`Any` is the *abstract* type of everything.
-
-Julia does have abstract types which are helpful for multiple dispatch. More about multiple dispatch later 😉
-"""
-
-# ╔═╡ 23d4ac67-05ec-4b3d-8368-86256076be62
-# The type hierarchy
-supertypes(Int64)
-
-# ╔═╡ 866edf5e-a76c-448d-98e8-925eaed5eba5
-subtypes(Number)
-
-# ╔═╡ a8ea4ac1-7f62-4485-a8c5-8ccf00c45720
-subtypes(Real)
-
-# ╔═╡ 786a96be-16cd-4f1b-9b5f-138e232d3183
-subtypes(Integer)
-
-# ╔═╡ dd8dad86-6bc2-4489-8469-7eac80fc41bb
-subtypes(Signed)
-
-# ╔═╡ c336d5f6-80ee-4994-a55f-2d6b3aa3d559
-# Hierarchy of Float64
-supertypes(Float64)
-
 # ╔═╡ 87b43f26-7437-4ee9-9b83-5b21e86dd0c9
 md"""
 ⚠️ Try to avoid using mixed arrays (type `Any`)!
@@ -440,20 +567,26 @@ typeof(first_vector[1:2])
 
 # ╔═╡ 605ee405-ec83-4064-adc8-861d95513e5e
 md"""
-### Views
+## Views
-Views are not a copy of an array, but a reference to a part of it.
+Views are not a copy of an array, but a *reference* to a part of it. It is best explained by an example:
 """
 
 # ╔═╡ 22a7baee-6533-43f7-8503-e5d5537a8c78
+# Don't panic!
+# You don't have to understand everything in this "long" piece of code.
+# It is only meant for concept explanation.
+# The output is important!
 begin
 	v = [3, 6, 9, 12]
 	@show v
-	
+
+	# Copy
 	@show copy_of_v = v[1:3]
 
+	# View
 	@show view_of_v = view(v, 1:3)
 
-	println()
+	println() # Generate a new empty line
 
 	new_value = 100
 	v[1] = new_value
@@ -484,42 +617,119 @@ begin
 	@show view_of_v
 end
 
+# ╔═╡ 90b0fb2f-eb2d-4d06-96da-a4605ce61c41
+md"""
+Using views is important for performance. Copying or initializing an array is expensive!
+
+This is because a free place in the memory has to be found and assigned to the new array. This process is called *memory allocation*. More about allocations and performance improvements in the days 😉
+"""
+
+# ╔═╡ 786682f6-692d-488d-8dab-231b0111d07f
+md"""
+## Vector operations
+"""
+
+# ╔═╡ 16a9cb53-2812-4ed3-afe4-96c0b116ad9a
+v1 = [1, 2, 3]
+
+# ╔═╡ dfd91d6b-65a5-454b-a0f9-6ed267def022
+v2 = [0, 5, 10]
+
+# ╔═╡ e68c54d8-3fb8-4aae-a334-665fdb8db1f0
+v1 + v2
+
+# ╔═╡ 8d74d994-3d4e-40ba-97cb-6dac1003fb8f
+# Now we have access to the function `dot` (and many others)
+# Dot product
+dot(v1, v2)
+
+# ╔═╡ af87251f-a37c-4088-8f4d-3803778bd97e
+# Cross product
+cross(v1, v2)
+
 # ╔═╡ 2ac5d431-1a4d-4db2-8954-97e011cd2175
 md"""
-### Matrices
+## Matrices
 """
 
 # ╔═╡ d30b3a5f-e14c-45ea-89a4-cf710733a2ee
 # Readable method to define a matrix
 first_matrix = [
-1 2
+	1 2
-3 4
+	3 4
 ]
 
 # ╔═╡ 6bb730e4-b5aa-4e7b-9ccd-9298db061e7f
 # Easier to write method
 second_matrix = [1 2; 3 4]
 
+# ╔═╡ 2ef34862-0578-41fe-adad-0e894c287dd5
+third_matrix = [
+	1 2 3
+	4 5 6
+]
+
+# ╔═╡ 09f2d0f9-cd0e-45e4-a159-cb360292dac1
+# Dimensions: n × m
+size(third_matrix)
+
+# ╔═╡ 6f6a875e-fe60-47ba-8837-60edef1b20e0
+# Determinant, from LinearAlgebra
+det(first_matrix)
+
+# ╔═╡ e99a89ab-af3a-42f5-b1c1-22e13a761eeb
+# Inverse matrix, from LinearAlgebra
+inv_first_matrix = inv(first_matrix)
+
+# ╔═╡ f45b7774-df6f-4019-9217-e88d99babdb3
+# Matrix multiplication
+inv_first_matrix * first_matrix
+
+# ╔═╡ db1b6e53-3116-48df-b098-1c3045be0dad
+# Calculate eigenvalues and vectors, from LinearAlgebra
+vals, vecs = eigen(first_matrix)
+
+# ╔═╡ cd5abd71-1bf8-484f-a46a-99cc8b994b91
+# Eigenvalues
+vals
+
+# ╔═╡ 7cef46dc-803a-4a7a-9663-148b6de4a267
+# First eigenvector
+vecs[:, 1]
+
+# ╔═╡ 70710989-9139-4970-a7b0-5702571e59a4
+# Second eigenvector
+vecs[:, 2]
+
+# ╔═╡ a9f39e34-4c2c-48f2-9353-babe1bc3cd05
+md"""
+## More dimensions
+
+One of the best ways to initialize arrays is to use `zeros`, `ones` or `fill` providing the dimensions.
+
+After initialization, you can populate the array (inplace).
+"""
+
 # ╔═╡ 83eca43d-2280-40f1-bf2a-016a843362a3
-first_tensor = zeros(3, 3, 3)
+# Tensor with the following dimensions: 3 × 3 × 3
-
+begin
-# ╔═╡ 64e47c03-356c-4a71-b463-4c247ea861cb
+	first_tensor = zeros(3, 3, 3)
-first_tensor[1, 1, 1] = 1.0
+	first_tensor[1, 1, 1] = 1.0
-
+	first_tensor[2, :, :] .= 4.0 # The dot is important! (Remember views)
-# ╔═╡ 23bdd308-3efa-49fd-9dcd-d4d1989383ee
+	first_tensor
-first_tensor[2, :, :] .= 4.0
+end
-
-# ╔═╡ 1a0b5604-fa4b-4291-baf0-97340aff9bca
-first_tensor
 
 # ╔═╡ f4c48701-d90e-48d6-bf9d-539c7fb7c7a5
-md"""
+ones(3, 2)
-Of course, you can have more than 3 dimensions!
+
-"""
+# ╔═╡ 875cb2c2-e78d-41e3-808b-c6948f215b76
+# Fill with a value other than 0 or 1
+fill(42, (2, 2, 3))
 
 # ╔═╡ 00000000-0000-0000-0000-000000000001
 PLUTO_PROJECT_TOML_CONTENTS = """
 [deps]
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
 
 [compat]
@@ -736,6 +946,7 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
 # ╟─2c5e32f4-1d7d-4494-b025-a90d17919756
 # ╟─21590bf1-1e1c-46b4-a2b6-7eb915e121ab
 # ╟─d04af0fd-5ced-4f4f-b157-dd170e2ef8c8
+# ╟─b559ef1a-76cf-4464-b5ec-f7d6bfa892e6
 # ╟─938adcfe-8d1b-4c77-8d82-c48415f5673e
 # ╠═73190799-fd03-4cc4-9b4e-c523bc310468
 # ╠═4c242a67-6445-48e7-a6c3-418a489b89ba
@@ -763,6 +974,17 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
 # ╠═204cf77f-bf37-4110-9c9f-1f9236301ba9
 # ╠═750bba32-e695-48f1-af70-70c94d13366b
 # ╠═0b663bcb-4ff4-4597-b28b-b58c9cbfa181
+# ╟─944e2d37-8280-47b8-b874-97221955d048
+# ╠═23d4ac67-05ec-4b3d-8368-86256076be62
+# ╟─e8d7de2f-7c7e-47cb-9364-27d583652167
+# ╠═866edf5e-a76c-448d-98e8-925eaed5eba5
+# ╠═a8ea4ac1-7f62-4485-a8c5-8ccf00c45720
+# ╠═786a96be-16cd-4f1b-9b5f-138e232d3183
+# ╠═dd8dad86-6bc2-4489-8469-7eac80fc41bb
+# ╠═fac04aa7-28e9-4f93-9312-a8f8f93c0877
+# ╠═86bc0ff0-b6bf-4700-a741-36323be58391
+# ╠═ef35a3a7-c1df-4952-aa41-1ed22d7f3981
+# ╠═c336d5f6-80ee-4994-a55f-2d6b3aa3d559
 # ╟─608d4433-6e68-4f95-8581-437234b58e87
 # ╠═beadbfd3-0015-449a-b6e7-b5182b396c1d
 # ╠═552fafa4-fad5-4efe-895f-255b3ec5c858
@@ -792,9 +1014,23 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
 # ╠═d2607457-1794-4a0f-af41-cb80aadb598f
 # ╠═23dbbe13-d997-4f9f-a300-7cb78c4fb8ee
 # ╠═e767971a-7e1d-4a78-88d7-03e4ae4d51db
+# ╟─3102810f-3467-4ed8-86c0-16e9177fa69d
+# ╠═c1c705d2-7e46-4811-9fb1-6b88b5a4140e
+# ╟─00456b15-5d1c-4c74-a875-31ff9c8e1789
+# ╠═91c4c623-5680-4b35-a694-2bd2612def94
+# ╠═2a85d95b-51d2-4ea0-a2a2-43307a725f2a
+# ╠═9a78bf14-7fb4-448a-a8dd-69e244a0a297
+# ╠═5281ef32-5de6-4488-8430-e5652cbf8299
+# ╠═eec41279-e038-4415-81b3-ad5d4c396011
+# ╠═d2333817-e941-429b-b8e3-2ff07669096b
+# ╠═be0ff87b-229a-433e-a49e-2f1ced5bb9aa
+# ╠═96803a1e-0779-4eab-b120-b5569a44ac7b
+# ╠═5ab233d2-f360-4362-b1f8-3f3ae2a4fee1
+# ╠═0d100501-de84-4a5c-beb7-8ff9e83c473d
 # ╟─9e3f698d-e57b-46c2-98e0-157fa7b06ae6
 # ╟─d1680205-a8eb-4ef6-ae4f-059e7a30f5c1
 # ╠═b54ace0e-8947-46a3-842a-05b5cbfc4e87
+# ╠═0d5bfd45-79da-435b-9a98-8ed996bbc7b4
 # ╠═8c47710f-1ed2-40fd-9290-374b498380e3
 # ╟─a03b46cc-1b26-44c2-b83d-884e3dbbe4fa
 # ╠═68658408-188e-43f7-ad74-251172dec0a8
@@ -813,27 +1049,36 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
 # ╠═65d3ddc2-36ed-4126-9211-e838ffc0d859
 # ╠═641e8c05-3e80-47e5-be77-91090a5f799a
 # ╠═2b9a5867-ca82-4a27-a700-bd0bd6c89bbe
-# ╟─944e2d37-8280-47b8-b874-97221955d048
-# ╠═23d4ac67-05ec-4b3d-8368-86256076be62
-# ╠═866edf5e-a76c-448d-98e8-925eaed5eba5
-# ╠═a8ea4ac1-7f62-4485-a8c5-8ccf00c45720
-# ╠═786a96be-16cd-4f1b-9b5f-138e232d3183
-# ╠═dd8dad86-6bc2-4489-8469-7eac80fc41bb
-# ╠═c336d5f6-80ee-4994-a55f-2d6b3aa3d559
 # ╟─87b43f26-7437-4ee9-9b83-5b21e86dd0c9
 # ╠═22f5ebc1-fd5c-4ee7-b169-8144fbd9b570
 # ╠═7479c420-2e04-4fe7-823c-3fde9efb54ca
 # ╠═4cd82256-be63-4db1-b2af-82a1358f4881
-# ╠═605ee405-ec83-4064-adc8-861d95513e5e
+# ╟─605ee405-ec83-4064-adc8-861d95513e5e
 # ╠═22a7baee-6533-43f7-8503-e5d5537a8c78
+# ╟─90b0fb2f-eb2d-4d06-96da-a4605ce61c41
+# ╟─786682f6-692d-488d-8dab-231b0111d07f
+# ╠═16a9cb53-2812-4ed3-afe4-96c0b116ad9a
+# ╠═dfd91d6b-65a5-454b-a0f9-6ed267def022
+# ╠═e68c54d8-3fb8-4aae-a334-665fdb8db1f0
+# ╠═1f347724-1db2-48f0-87df-4e63ad6e8820
+# ╠═8d74d994-3d4e-40ba-97cb-6dac1003fb8f
+# ╠═af87251f-a37c-4088-8f4d-3803778bd97e
 # ╟─2ac5d431-1a4d-4db2-8954-97e011cd2175
 # ╠═d30b3a5f-e14c-45ea-89a4-cf710733a2ee
 # ╠═6bb730e4-b5aa-4e7b-9ccd-9298db061e7f
+# ╠═2ef34862-0578-41fe-adad-0e894c287dd5
+# ╠═09f2d0f9-cd0e-45e4-a159-cb360292dac1
+# ╠═6f6a875e-fe60-47ba-8837-60edef1b20e0
+# ╠═e99a89ab-af3a-42f5-b1c1-22e13a761eeb
+# ╠═f45b7774-df6f-4019-9217-e88d99babdb3
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