Notebook 3 – Math 2121, Fall 2021

In today's notebook we'll try to get some feeling for vectors in $R^{2}$ and $R^{3}$ and their linear combinations. We'll also think about what a solution to a vector equation means. The code here involves a lot of new plotting functions in Julia.

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md"# Notebook 3 -- Math 2121, Fall 2021
In today's notebook we'll try to get some feeling for vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$ and their linear combinations. We'll also think about what a solution to a vector equation means. The code here involves a lot of new plotting functions in Julia."

17.2 μs

Running this notebook (optional)

If you have Pluto up and running, you can access the notebook we are currently viewing by entering this link (right click -> Copy Link) in the Open from file menu in Pluto.

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md"## Running *this* notebook (optional)
If you have Pluto up and running, you can access the notebook we are currently viewing by entering [this link](http://www.math.ust.hk/~emarberg/teaching/2021/Math2121/julia/03_Math2121_Fall2021.jl) (right click -> Copy Link)  in the *Open from file* menu in Pluto.
"

1.7 ms

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begin
    using PlutoUI
    using Plots
end

457 μs

Vectors in $R^{2}$

Let's start out by drawing some vectors in $R^{2}$ .

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md"## Vectors in $\mathbb{R}^2$
​
Let's start out by drawing some vectors in $\mathbb{R}^2$."

8.7 μs

Create some vectors in $R^{2}$

u1 = v1 = w1 =

u2 = v2 = w2 =

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begin   
    u1_slider = @bind u1 Slider(-10:0.1:10, default=4, show_value=false)
    u2_slider = @bind u2 Slider(-10:0.1:10, default=8, show_value=false)
​
    v1_slider = @bind v1 Slider(-10:0.1:10, default=-10, show_value=false)
    v2_slider = @bind v2 Slider(-10:0.1:10, default=6, show_value=false)
​
    w1_slider = @bind w1 Slider(-10:0.1:10, default=2, show_value=false)
    w2_slider = @bind w2 Slider(-10:0.1:10, default=-9, show_value=false)
    
    
    md"""**Create some vectors in $$\mathbb{R}^2$$**
    
    `u1` = $(u1_slider) `v1` = $(v1_slider) `w1` = $(w1_slider)
    
    `u2` = $(u2_slider) `v2` = $(v2_slider) `w2` = $(w2_slider)
    
    """
end

131 ms

The vectors we created:

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md"The vectors we created:"

7.9 μs

u = [   4]     v = [ -10]     w = [   2]
    [   8]         [   6]         [  -9]

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begin
    s = [string(u1), string(v1), string(w1), string(u2), string(v2), string(w2)]
    pad = maximum([4, maximum(map(length,s))])
    for i=1:length(s)
        s[i] = " "^(pad - length(s[i])) * s[i]
    end
    
    top = join(["u = [$(s[1])]", "v = [$(s[2])]", "w = [$(s[3])]"], "     ")
    bot = join(["    [$(s[4])]", "    [$(s[5])]", "    [$(s[6])]"], "     ")
    Text(join([top, bot], "\n"))    
end

3.1 ms

Display: u v w

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md"""**Display**: 
`u` $(@bind uon html"<input type=checkbox >") `v` $(@bind von html"<input type=checkbox >") `w` $(@bind won html"<input type=checkbox >")
"""

13.9 ms

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begin
    lims = (
        -(maximum([1, uon + von + won])) * 10, 
        maximum([1, uon + von + won]) * 10
    )
    
    p1 = scatter([0], [0], xlims = lims, ylims = lims, 
        legend = false, label = "origin",  aspect_ratio=:equal,
        title = "Vectors u, v, w")
    
    if uon
        quiver!(quiver = ([u1],[u2]), [0], [0], color = [:blue],)
    end
    
    if von
        quiver!(quiver = ([v1],[v2]), [0], [0], color = [:orange],)
    end
    
    if won
        quiver!(quiver = ([w1],[w2]), [0], [0], color = [:green],)
    end 
    
    p2 = scatter([0], [0], xlims = lims, ylims = lims, 
        legend = false, label = "origin",  aspect_ratio=:equal,
        title = "Sum of vectors")
    
    quiver!(
        quiver = ([uon*u1 + von*v1 + won*w1],[uon*u2 + von*v2 + won*w2]), 
        [0], [0], color = [:black],
    )
    
    quiver!(
        quiver = ([uon*u1],[uon*u2]), 
        [0], [0], 
        color = [:blue], 
        linestyle = :dashdot,
    )
    quiver!(
        quiver = ([von*v1],[von*v2]), 
        [uon*u1], [uon*u2], 
        color = [:orange], 
        linestyle = :dashdot,
    )
    quiver!(
        quiver = ([won*w1],[won*w2]), 
        [uon*u1 + von*v1], [uon*u2 + von*v2], 
        color = [:green], 
        linestyle = :dashdot,
    )
    
    plot(p1, p2, layout=2)
end

4.0 ms

The vector $u$ is shown in blue, $v$ in orange, and $w$ in green.

The right hand graph shows the sum of the displayed vectors on the left as the solid arrow. The dotted arrows are just copies of u, v, and w moved around.

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md"The vector $u$ is shown in blue, $v$ in orange, and $w$ in green.
​
The right hand graph shows the sum of the displayed vectors on the left as the solid arrow. The dotted arrows are just copies of u, v, and w moved around."

10.2 μs

Linear combinations

Now that we have chosen some vectors $u$ , $v$ , $w$ , let's think about the meaning of the linear combination $a u + b v + c w$ where $a, b, c \in R$ are scalars.

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md"## Linear combinations
​
Now that we have chosen some vectors $u$, $v$, $w$, let's think about the meaning of the linear combination $au + bv + cw$ where $a,b,c \in \mathbb{R}$ are scalars."

23.3 μs

Choose some scalars in $R$

a = b = c =

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begin   
    a_slider = @bind a Slider(-10:0.1:10, default=1*uon, show_value=false)
    b_slider = @bind b Slider(-10:0.1:10, default=1*von, show_value=false)  
    c_slider = @bind c Slider(-10:0.1:10, default=1*won, show_value=false)
    
    md"""**Choose some scalars in $$\mathbb{R}$$**
    
    `a` = $(a_slider) `b` = $(b_slider) `c` = $(c_slider)
    """
end

78.0 μs

The scalars we chose:

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md"The scalars we chose:"

4.6 μs

a = 1     b = -1.4     c = 0

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Text(join([string("a = ", a), string("b = ", b), string("c = ", c)], "     "))

9.6 μs

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begin
    p3 = scatter([0], [0], xlims = lims, ylims = lims, 
        legend = false, label = "origin",  aspect_ratio=:equal,
        title = "a•u, b•v, and c•w")
    
    if uon
        quiver!(quiver = ([a*u1],[a*u2]), [0], [0], color = [:blue],)
    end
    
    if von
        quiver!(quiver = ([b*v1],[b*v2]), [0], [0], color = [:orange],)
    end
    
    if won
        quiver!(quiver = ([c*w1],[c*w2]), [0], [0], color = [:green],)
    end 
    
    p4 = scatter([0], [0], xlims = lims, ylims = lims, 
        legend = false, label = "origin",  aspect_ratio=:equal,
        title = "a•u + b•v + c•w")
    
    quiver!(
        quiver = (
            [a*uon*u1 + b*von*v1 + c*won*w1],
            [a*uon*u2 + b*von*v2 + c*won*w2]
        ), 
        [0], [0], color = [:black],
    )
    
    quiver!(
        quiver = ([a*uon*u1],[a*uon*u2]), 
        [0], [0], 
        color = [:blue], 
        linestyle = :dashdot,
    )
    quiver!(
        quiver = ([b*von*v1],[b*von*v2]), 
        [a*uon*u1], [a*uon*u2], 
        color = [:orange], 
        linestyle = :dashdot,
    )
    quiver!(
        quiver = ([c*won*w1],[c*won*w2]), 
        [a*uon*u1 + b*von*v1], [a*uon*u2 + b*von*v2], 
        color = [:green], 
        linestyle = :dashdot,
    )
    
    plot(p3, p4, layout=2)
end

5.8 ms

Varying $a$ among positive values scales the length of the arrow that represents the vector $u$ .

If $a$ is negative then this scalaing reverses the direction of $u$ .

The same thing happens to the arrows representing $v$ and $w$ when we change $b$ and $c$ .

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md"Varying $a$ among positive values scales the length of the arrow that represents the vector $u$. 
​
If $a$ is negative then this scalaing reverses the direction of $u$. 
​
The same thing happens to the arrows representing $v$ and $w$ when we change $b$ and $c$."

7.7 μs

Vector equations

Let's now think of $a, b, c$ as the variables $x_{1} = a$ and $x_{2} = b$ and $x_{3} = c$ .

Choose a target vector $t = [\begin{array}{l} t_{1} \\ t_{2} \end{array}] \in R^{2}$ .

What does a solution to the equation $x_{1} u + x_{2} v + x_{3} w = t$ mean?

The solutions $x = [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}]$ to this equation are the same as the solutions to the linear system whose augmented matrix is $A = [\begin{array}{cccc} u_{1} & v_{1} & w_{1} & t_{1} \\ u_{2} & v_{2} & w_{2} & t_{2} \end{array}]$ .

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md"## Vector equations
​
Let's now think of $a,b,c$ as the variables $x_1=a$ and $x_2=b$ and $x_3=c$.
​
Choose a target vector $t =\left[\begin{array}{l} t_1 \\ t_2\end{array}\right] \in \mathbb{R}^2$.
​
What does a solution to the equation $x_1 u + x_2 v + x_3 w = t$ mean?
​
The solutions $x = \left[\begin{array}{l} x_1 \\ x_2 \\ x_3\end{array}\right]$ to this equation are the same as the solutions to the linear system whose augmented matrix is 
$$A = \left[\begin{array}{ccc|c} u_1 & v_1 & w_1 & t_1 \\ u_2 & v_2 & w_2 & t_2 \end{array}\right]$$."

8.5 μs

Let's create a random target vector $t$ .

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md"Let's create a random target vector $t$."

5.2 μs

target

2×1 Matrix{Float64}:
 -19.6
  11.3

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target = rand(-20:0.1:20, 2, 1)

5.6 μs

Here's our coefficient matrix.

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md"Here's our coefficient matrix."

5.9 μs

coefficient_matrix

2×3 Matrix{Int64}:
 4  -10   2
 8    6  -9

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coefficient_matrix = [u1 v1 w1; u2 v2 w2]

27.3 μs