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Rmd 58aaac2 Dave Tang 2024-04-21 Paul Dirac’s notation
html c578a55 Dave Tang 2024-04-21 Build site.
Rmd 2442c92 Dave Tang 2024-04-21 Another perpendicular example
html 3d298bc Dave Tang 2024-04-21 Build site.
Rmd ed752d3 Dave Tang 2024-04-21 Orthogonal means perpendicular
html 75b63ae Dave Tang 2024-04-20 Build site.
Rmd fb1ef5b Dave Tang 2024-04-20 Parallelogram law for vector addition
html 8ae3753 Dave Tang 2024-04-20 Build site.
Rmd 97fe3ef Dave Tang 2024-04-20 Fix formula
html 96459ae Dave Tang 2024-04-20 Build site.
Rmd 8d79433 Dave Tang 2024-04-20 Expand and rearrange vectors section
html cd09128 Dave Tang 2024-04-20 Build site.
Rmd 1a7c3c1 Dave Tang 2024-04-20 Linear algebra basics

Paul Dirac’s notation

How to denote Bra-ket notation in LaTeX?

\[ \langle a | b \rangle \\ \langle p \vert q \rangle \\ \langle a \mid b \rangle \\ \langle \frac{m}{2} \vert \frac{n}{2} \rangle \]

Vectors

A vector is just a list of numbers and the dimension of the vector is the number of numbers in the list. Vectors can be written vertically (column vectors) or horizontally (row vectors). The numbers that make up a vector are often called entries.

A simple five-dimensional (column) vector.

\[ \vec{x} = \begin{pmatrix}8\\6\\7\\5\\3\end{pmatrix} \]

The length of a vector is the distance from its initial point to its terminal point. Calculating vector length is the square root of the sum of squares.

\[ {\lvert}\vec{v}{\rvert} = \sqrt{ \sum_{i=1}^{n}{v_i^2} } \]

Length of the \((3, 1)\) vector.

\[ \sqrt{3^2 + 1^2} = \sqrt{10} \]

A normal vector (or unit vector) is a vector of length 1. For example, the vector \((2/5, 4/5, 1/5, 2/5)\) has length 1.

\[ {\lvert}\vec{u}{\rvert} = \sqrt{(\frac{2}{5})^2 + (\frac{4}{5})^2 + (\frac{1}{5})^2 + (\frac{2}{5})^2} = \sqrt{\frac{4}{25} + \frac{16}{25} + \frac{1}{25} + \frac{4}{25}} = \sqrt{\frac{25}{25}} = 1 \]

We can multiply a vector by a number and in linear algebra, numbers are often called scalars. Scalar multiplication just refers to multiplying by a number. We do this by multiplying each of the entries by the scalar.

\[ \vec{v} = [3, 6, 8, 4] \times 1.5 = [4.5, 9, 12, 6] \]

If we multiply a vector by the reciprocal of its length, we obtain a unit vector. For example the vector \((3, 1)\) has length \(\sqrt{10}\).

\[ \frac{1}{\sqrt{10}} \times \begin{pmatrix}3\\1\end{pmatrix} = \begin{pmatrix} \frac{3}{\sqrt{10}}\\\frac{1}{\sqrt{10}}\end{pmatrix} \\ \sqrt{\bigg(\frac{3}{\sqrt{10}}\bigg)^2 + \bigg(\frac{1}{\sqrt{10}}\bigg)^2} = \sqrt{\frac{9}{10} + \frac{1}{10}} = \sqrt{1} = 1 \]

The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space.

\[ (\vec{x}, \vec{y}) = \vec{x} \cdot \vec{y} = \sum_{i=1}^{n}{x_{i}y_{i}} \]

Given two vectors with the same dimension, we can add them together to get a new vector with the same dimension. The first entry of the new vector is just the addition of the first entries of the two vectors and so on.

\[ A = [a_{1}, a_{2}, \dotsc, a_{n}] \\ B = [b_{1}, b_{2}, \dotsc, b_{n}] \\ A + B = [a_{1} + b_{1}, a_{2} + b_{2}, \dotsc, a_{n} + b_{n}] \]

Vector addition can be pictured by what is often called the parallelogram law for vector addition. If the vector \(B = (1, 2)\) is drawn so that its initial point is at the terminal point of \(A = (3, 1)\), then the vector that goes from the initial point of \(A\) to the terminal point of \(B\) is \(A + B\). This can be drawn giving a triangle.

plot_vec_add <- function(my_df){
  my_df <- rbind(my_df, c(sum(my_df$x), sum(my_df$y)))
  my_arrow <- arrow(length = unit(0.03, "npc"), type = "closed")
  
  ggplot(my_df, aes(x, y)) +
    geom_point() +
    scale_x_continuous(limits = c(0, max(my_df[, 1]))) +
    scale_y_continuous(limits = c(0, max(my_df[, 2]))) +
    geom_segment(
      aes(
        x=0,
        y=0,
        xend=my_df[1,1],
        yend=my_df[1,2]
      ),
      lty=3,
      arrow = my_arrow
    ) +
    geom_segment(
      aes(
        x=0,
        y=0,
        xend=my_df[2,1],
        yend=my_df[2,2]
      ),
      lty=3,
      arrow = my_arrow
    ) +
    geom_segment(
      aes(
        x=my_df[1,1],
        y=my_df[1,2],
        xend=my_df[3,1],
        yend=my_df[3,2]
      ),
      lty=3,
      arrow = my_arrow
    ) +
    geom_segment(
      aes(
        x=my_df[2,1],
        y=my_df[2,2],
        xend=my_df[3,1],
        yend=my_df[3,2]
      ),
      lty=3,
      arrow = my_arrow
    ) +
    geom_segment(
      aes(
        x=0,
        y=0,
        xend=my_df[3,1],
        yend=my_df[3,2]
      ),
      lty=1,
      arrow = my_arrow
    ) +
    theme_minimal() +
    NULL
}

my_df <- data.frame(
  x = c(3, 1),
  y = c(1, 2)
)

plot_vec_add(my_df)
Parallelogram law for vector addition

Parallelogram law for vector addition

Version Author Date
3d298bc Dave Tang 2024-04-21

Orthogonality

The parallelogram law for vector addition figure helps us visualise some basic properties of vector addition. One of the most important comes from the Pythagorean theorem. We know that if \(a\), \(b\), and \(c\) represent the lengths of the three sides of a triangle, then \(a^2 + b^2 = c^2\) if and only if the triangle is a right triangle. The figure then tells us that two vectors \(a\) and \(b\) are perpendicular **if and only if and only if \(len(a)^2 + len(b)^2 = len(a+b)^2\).

my_df <- data.frame(
  x = c(0, 4),
  y = c(4, 0)
)
plot_vec_add(my_df)

Version Author Date
3d298bc Dave Tang 2024-04-21

Another example.

my_df <- data.frame(
  x = c(1, 5),
  y = c(5, -1)
)
plot_vec_add(my_df)

Version Author Date
c578a55 Dave Tang 2024-04-21

The word orthogonal means exactly the same thing as perpendicular and it is the word used in linear algebra. Two vectors \(a\) and \(b\) are orthogonal if and only if \(len(a)^2 + len(b)^2 = len(a+b)^2\).

Two vectors are orthogonal if their inner product is zero. Using the example of the plot above where \(a = (1, 5)\) and \(b = (5, -1)\).

\[ [1, 5] \cdot [5, -1] = 1(5) + 5(-1) = 0 \]

Another example.

\[ [2, 1, -2, 4] \cdot [3, -6, 4, 2] = 2(3) + 1(-6) - 2(4) + 4(2) = 0 \]

Vectors of unit length that are orthogonal to each other are said to be orthonormal.

\[ \vec{u} = [2/5, 1/5, -2/5, 4/5] \\ \vec{v} = [3 / \sqrt{65}, -6 / \sqrt{65}, 4 / \sqrt{65}, 2 / \sqrt{65}] \\ {\lvert}\vec{u}\rvert = \sqrt{(2/5)^2 + (1/5)^2 + (-2/5)^2 + (4/5)^2} = 1 \\ {\lvert}\vec{v}\rvert = \sqrt{(3 / \sqrt{65})^2 + (-6 / \sqrt{65})^2 + (4 / \sqrt{65})^2 + (2 / \sqrt{65})^2} = 1 \\ \vec{u} \cdot \vec{v} = \frac{6}{5\sqrt{65}} - \frac{6}{5\sqrt{65}} - \frac{8}{5\sqrt{65}} + \frac{8}{5\sqrt{65}} = 0 \]

If we have row and column vectors with the same dimension, we can multiply them (row on the left and column on the right) to obtain a number.

\[ [a_1, a_2, \cdots, a_n] \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} = a_1b_1 + a_2b_2 + \cdots + a_nb_n \]

Matrices

Matrix of numbers.

\[ \begin{bmatrix} 17 & 18 & 5 & 5 & 45 & 1 \\ 42 & 28 & 30 & 15 & 115 & 3 \\ 10 & 10 & 10 & 21 & 51 & 2 \\ 28 & 5 & 65 & 39 & 132 & 5 \\ 24 & 26 & 45 & 21 & 116 & 4 \end{bmatrix} \]

Matrix with subscripts and dots

\[ A = \begin{bmatrix} a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \end{bmatrix} \]

Square matrix

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]

Transpose.

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \\ A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]

Matrix multiplication.

\[ AB = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ -1 & 4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 9 & 16 \\ 0 & 26 \end{bmatrix} \]

Identity matrix.

\[ AI = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \]

Orthogonal matrix.

\[ AA^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & 4/5 \\ 0 & -4/5 & 3/5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Diagonal matrix

\[ A = \begin{bmatrix} a_{11} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & 0 \\ 0 & 0 & 0 & a_{mm} \end{bmatrix} \]

Determinant of a 2x2 matrix.

\[ {\lvert}A\rvert = \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc \]


sessionInfo()
R version 4.3.3 (2024-02-29)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 22.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so;  LAPACK version 3.10.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

time zone: Etc/UTC
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] lubridate_1.9.3 forcats_1.0.0   stringr_1.5.1   dplyr_1.1.4    
 [5] purrr_1.0.2     readr_2.1.5     tidyr_1.3.1     tibble_3.2.1   
 [9] ggplot2_3.5.0   tidyverse_2.0.0 workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    stringi_1.8.3    
 [5] hms_1.1.3         digest_0.6.35     magrittr_2.0.3    timechange_0.3.0 
 [9] evaluate_0.23     grid_4.3.3        fastmap_1.1.1     rprojroot_2.0.4  
[13] jsonlite_1.8.8    processx_3.8.4    whisker_0.4.1     ps_1.7.6         
[17] promises_1.3.0    httr_1.4.7        fansi_1.0.6       scales_1.3.0     
[21] jquerylib_0.1.4   cli_3.6.2         rlang_1.1.3       munsell_0.5.1    
[25] withr_3.0.0       cachem_1.0.8      yaml_2.3.8        tools_4.3.3      
[29] tzdb_0.4.0        colorspace_2.1-0  httpuv_1.6.15     vctrs_0.6.5      
[33] R6_2.5.1          lifecycle_1.0.4   git2r_0.33.0      fs_1.6.3         
[37] pkgconfig_2.0.3   callr_3.7.6       pillar_1.9.0      bslib_0.7.0      
[41] later_1.3.2       gtable_0.3.4      glue_1.7.0        Rcpp_1.0.12      
[45] highr_0.10        xfun_0.43         tidyselect_1.2.1  rstudioapi_0.16.0
[49] knitr_1.46        farver_2.1.1      htmltools_0.5.8.1 labeling_0.4.3   
[53] rmarkdown_2.26    compiler_4.3.3    getPass_0.2-4