Notebook 6 – Math 2121, Fall 2020

md"# Notebook 6 -- Math 2121, Fall 2020

In today's notebook we'll first explore some more plots of linear transformations. Then we'll take a look at the frequency that a random linear transformation is one-to-one or onto.

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##### 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/2020/Math2121/julia/06_Math2121_Fall2020.jl) (right click -> Copy Link)  in the *Open from file* menu in Pluto."

​

begin

    r_slider = @bind r Slider(0:0.1:2 * pi, default=0, show_value=false)

    theta_slider = @bind theta Slider(0:0.1:2 * pi, default=0, show_value=true)

    x1_slider = @bind x1 Slider(-2:0.1:2, default=1, show_value=false)

    x2_slider = @bind x2 Slider(-2:0.1:2, default=0, show_value=false)

​

    y1_slider = @bind y1 Slider(-2:0.1:2, default=0, show_value=false)

    y2_slider = @bind y2 Slider(-2:0.1:2, default=1, show_value=false)

    k_slider = @bind k Slider(-1:0.1:1, default=1, show_value=true)

    md"""**input parameters**

    `x1` = $(x1_slider) `y1` = $(y1_slider) rotate = $(r_slider)

    `x2` = $(x2_slider) `y2` = $(y2_slider) 

    **matrix parameters**

    `θ` = $(theta_slider)  `k` = $(k_slider)

    """

end

begin

    rmat = [cos(r) -sin(r); sin(r) cos(r)]

    x = rmat * [x1; x2]

    y = rmat * [y1; y2]

    pprint("x", "y", x, y)

end

begin

    p1 = input_plot(x, y, "x and y")

    p2 = transformation_plot(A, x, y, "Ax and Ay")

    plot(p1, p2, layout=2)

end

md"##### Random matrices, onto and one-to-one linear transformations

​

We can generate a random linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$

by generating a random $m\times n$ matrix $A$ and setting $T(x) = Ax$.

​

If $n > m$ then $T$ is never one-to-one, and if $n < m$ then $T$ is never onto.

If $n \leq m$, however, then a random linear transformation $T$ is injective with a high probability. 

If $n \geq m$, similarly, then a random linear transformation $T$ is surjective with a high probability.

​

In other words, a sufficiently random linear transformation is almost always one-to-one if this is possible (that is, if $n \leq m$) and almost always onto if this is possible (that is, if $n \geq m$).

​

If $n=m$ then a random linear transformation is almost always both one-to-one and onto (which will be our definition of an *invertible function* next lecture).

​

To explore this phenomenon, we'll consider our usual model of random matrix which is not *that* random: namely, $01$-matrices whose entries are independently $0$ or $1$ with some probability $p$. For such matrices, we can see some interesting behavior while also observing convergence to the ''almost always'' properties described above.

"

md"##### Helper methods"

plotly()

md"##### Parameters"

begin

    M_slider = @bind M Slider(1:50, default=2, show_value=true)

    N_slider = @bind N Slider(1:50, default=2, show_value=true)

    md"""

    `nrows` = $(M_slider) 

    `ncols` = $(N_slider)

    """

end

md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size 

**$(M) by $(N)**. For each p, we compute the proportition of these matrices that correspond to **onto** linear transformations, as well as the standard deviation of this statistic. 

​

Then we plot both as a function of p."

begin

    onto_title = "Random 01-matrices, size $(M)-by-$(N): onto transformations"

    accumulate_plot(is_onto, onto_title, trials, M, N)

end

md"Interesting properties of this graph: when $p$ is not too small or too large, our model of a random matrix is ''sufficiently'' random, so the measured proportion is 1.0: almost every random matrix corresponds to an onto linear transformation.

​

The standard deviation is always bimodal, with maximum value 0.5 attained at two values $p_1$ and $p_2$ of the probability $p$, which are also where the mean and standard deviation graphs intersect. 

​

What are these two values of $p$? They are not symmetric, that is, $p_2 \neq 1-p_1$."

md"Why does the mean have value $0$ when $p=0$ and when $p=1$?"

md"Here is a random 01-matrix for $p=0$:"

md"Here is a random 01-matrix for $p=1$:"

md"In both cases the number of pivots is much less than the number of rows or columns, so these matrices correspond to linear transformations that are neither onto nor one-to-one."

md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size 

**$(N) by $(M)** (note: reversed from above). For each p, we compute the proportition of these matrices that correspond to **one-to-one** linear transformations, as well as the standard deviation of this statistic. 

​

Then we plot both as a function of p."

begin

    one_to_one_title = "Random 01-matrices, size $(N)-by-$(M): 1-to-1 transformations"

    accumulate_plot(is_one_to_one, one_to_one_title, trials, N, M)

end

md"Why are two graphs nearly the same?

​

The **transpose** of a matrix $A$ is then matrix $A^T$ formed by interchanging the rows and columns:

​

$$\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]^T

=\left[\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right].$$

​

We'll see later in the class, that the number of pivot positions in $A$ is the same as in $A^T$."

md"This means that the number of $(M) by $(N) matrices with pivots in every row (corresponding to **onto** linear transformations) is the same as the number of $(N) by $(M) matrices with pivots in every column (corresponding to **one-to-one** linear transformations)."