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secao13Pluto.jl
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secao13Pluto.jl
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### A Pluto.jl notebook ###
# v0.19.9
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 39ab2159-9f73-477e-8086-2299e55170fc
using Plots, PlutoUI
# ╔═╡ f4c4a330-a259-11ed-04bc-a1ab9f48e7c9
md"""# Bases and Dimension
Given a space $\mathcal{E}$, it may be quite useful to be able to
address (or to access) each and every element of $\mathcal{E}$ using a
set of unequivocal coordinates. Latitudes and longitudes provide such
coordinates for geographical positioning on earth’s surface. For a more
generic space, we may need to define a basis for the space in more
appropriate terms.
Given a space $\mathcal{E}$ defined over a field $\mathbb{F}$, a set of
elements $\{\vbf_i\}\subseteq\mathcal{E}$, $i=1,\ 2,\ \ldots,\ r$, is a
*basis* of $\mathcal{E}$ if the elements $\vbf_i\in\{\vbf_i\}$ are
linearly independent and any element $\ubf\in\mathcal{E}$ can be written
as a linear combination of $\vbf_i$. Another way to put it is to say
that the elements $\vbf_i$, $i=1,\ 2,\ \ldots,\ r$ span the whole space
$\mathcal{E}$ and no other set of elements with less than $r$ elements
spans the space $\mathcal{E}$.
As many different coordinate systems may prove equally accurate for
determining position on the earth’s surface, many different bases may
span a given vector space.
::: example <span id="ex:canonic_basis" label="ex:canonic_basis"></span>
Let us investigate the vector space $\mathbb{R}^2$. As previously
defined, this space can be described as the set
$$\mathbb{R}^2=\{\vbf;\ \vbf=\left[v_1\ v_2\right]^\transp,\ v_1,\ v_2 \in \mathbb{R}\}$$
We may verify that any vector $\vbf\in\mathbb{R}^2$ can be written as a
linear combination of vectors $\ebf_1$ and $\ebf_2$ defined below:
$$\ebf_1=\left[\begin{matrix}1\\0\end{matrix}\right],\quad
\ebf_2=\left[\begin{matrix}0\\1\end{matrix}\right].$$ It is also
straightforward to verify that $\ebf_1,\ \ebf_2\in\mathbb{R}^2$.
Therefore the set of vectors $\{\ebf_1,\ \ebf_2\}$ span $\mathbb{R}^2$.
This particular set of vectors forms the canonic basis of
$\mathbb{R}^2$, although it may not be so easy, at least at this moment,
to ascertain that no set with less than two elements span the whole
space $\mathbb{R}^2$. Many other sets with two vectors can form a basis
of $\mathbb{R}^2$. We will see further in the chapter how to change from
one basis to another.
::: example <span id="ex:V_space_line" label="ex:V_space_line"></span>
Another interesting example we may construct is for a vector space
$\mathcal{V}\subset\mathbb{R}^2$ defined as
$$\mathcal{V}=\{\vbf;\ \vbf=\alpha \left[1 \ 2\right]^\transp,\ \alpha\in\mathbb{R}\}$$
In this example, $\mathcal{V}$ is a line, as seen in
Figure <a href="#fig:V_space_line" data-reference-type="ref"
data-reference="fig:V_space_line">2</a>. Although we can describe any
point along this line using any basis of $\mathbb{R}^2$ (after all,
$\mathcal{V}\subset\mathbb{R}^2$), we can certainly be more economical.
In fact, any vector $\vbf=\alpha\left[1 \ 2\right]^\transp$ is itself a
basis of $\mathcal{V}$.
The definition of basis implies that for every vector space there is a
set of vectors forming its basis. We have seen above that the number of
vectors in the basis is two for $\mathbb{R}^2$ and one for $\mathcal{V}$
in Example <a href="#ex:V_space_line" data-reference-type="ref"
data-reference="ex:V_space_line">[ex:V_space_line]</a>. This number
denotes the dimension of these vector spaces.
Given a vector space $\mathcal{U}$, we define the *dimension* of the
vector space as the total number of vectors in its basis. A given vector
space may have infinite dimension, or it may have dimension equal to
zero."""
# ╔═╡ 608f50aa-4ca5-42d0-9dfa-66335ac193eb
md"""Base $\begin{bmatrix}
1\\
0
\end{bmatrix}$ ,
$\begin{bmatrix}
0\\
1
\end{bmatrix}$"""
# ╔═╡ 230072e9-1634-48a3-8f6e-7194af144b48
begin
vertical = @bind vetorVertical html"<input type=range>"
horizontal = @bind vetorHorizontal html"<input type=range>"
md"""
Horizontal: $(horizontal)
Vertical: $(vertical)
"""
end
# ╔═╡ 96a5edcf-f533-48fc-9d86-53d4c8d51083
begin
plt=plot()
x = [vetorHorizontal, 0, vetorHorizontal]
y = [0, vetorVertical, vetorVertical]
scatter!(plt,x,y)
end
# ╔═╡ 0a9a2026-0a2b-4ec7-bb5c-b7c22da8b9f3
begin
verticalText = @bind vetorVerticalT html"<input type=text>"
horizontalText = @bind vetorHorizontalT html"<input type=text>"
md"""
Horizontal: $(horizontalText)
Vertical: $(verticalText)
"""
end
# ╔═╡ eb3baf23-d615-4d2d-844d-a5342ac39a10
try
verticalConvertido = parse(Float64, vetorVerticalT)
horizontalConvertido = parse(Float64, vetorHorizontalT)
begin
plt1=plot()
plot!(plt1,[0,horizontalConvertido],[0,verticalConvertido],arrow=true,color=:blue,linewidth=2,label="")
plot!(plt1,[0,0],[0,verticalConvertido],arrow=true,color=:black,linewidth=2,label="")
plot!(plt1,[0,horizontalConvertido],[0,0],arrow=true,color=:black,linewidth=2,label="")
end
catch
println("Insira um número real.")
end
# ╔═╡ d85c6f40-7203-41a3-a709-17878f7b938b
md"""Base $\begin{bmatrix}
1\\
2
\end{bmatrix}$ ,
$\begin{bmatrix}
4\\
3
\end{bmatrix}$"""
# ╔═╡ 47647cf5-738f-4a78-ad39-1fda5c282b33
begin
PrimeiroText = @bind primeirovetorT html"<input type=text>"
SegundoText = @bind segundovetorT html"<input type=text>"
md"""
Escalar do primeiro vetor: $(PrimeiroText)
Escalar do segundo vetor: $(SegundoText)
"""
end
# ╔═╡ 24d63613-d960-4e36-9522-4d2350c1af3c
md""" {{primeirovetorT}}$\begin{bmatrix}
1\\
2
\end{bmatrix}$ +
{{segundovetorT}}$\begin{bmatrix}
4\\
3
\end{bmatrix}$"""
# ╔═╡ 8a157655-7cc9-4eb8-8d10-934c4e4f38b3
try
primeiroConvertido = parse(Float64, primeirovetorT)
segundoConvertido = parse(Float64, segundovetorT)
begin
plt2 = plot()
plot!(plt2, [0,primeiroConvertido], [0, 2*primeiroConvertido], arrow=true, color=:black, linewidth=2, label="")
plot!(plt2, [0, 4*segundoConvertido],[0, 3*segundoConvertido], arrow=true, color=:black,linewidth=2,label="")
plot!(plt2, [0, primeiroConvertido + 4*segundoConvertido],[0, 2*primeiroConvertido + 3*segundoConvertido], arrow=true, color=:blue, linewidth=2, label="")
end
catch
println("Insira um número real.")
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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[compat]
Plots = "~1.38.3"
PlutoUI = "~0.7.49"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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