Linear algebra basics
2024-04-20
Last updated: 2024-04-20
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| Rmd | fb1ef5b | Dave Tang | 2024-04-20 | Parallelogram law for vector addition |
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| Rmd | 1a7c3c1 | Dave Tang | 2024-04-20 | Linear algebra basics |
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.
my_df <- data.frame(
x = c(3, 1),
y = c(1, 2)
)
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
Orthogonality
Two vectors are orthogonal if their inner product is zero.
\[ [2, 1, -2, 4] \cdot [3, -6, 4, 2] = 2(3) + 1(-6) - 2(4) + 4(2) = 0 \]
Vectors of unit length that are orthogonal to each other are said to be orthonormal.
\[ \vec{u} = [2/5, 1/5, -2/5, 4/5] \\ \vec{v} = [3 / \sqrt{65}, -6 / \sqrt{65}, 4 / \sqrt{65}, 2 / \sqrt{65}] \\ {\lvert}\vec{u}\rvert = \sqrt{(2/5)^2 + (1/5)^2 + (-2/5)^2 + (4/5)^2} = 1 \\ {\lvert}\vec{v}\rvert = \sqrt{(3 / \sqrt{65})^2 + (-6 / \sqrt{65})^2 + (4 / \sqrt{65})^2 + (2 / \sqrt{65})^2} = 1 \\ \vec{u} \cdot \vec{v} = \frac{6}{5\sqrt{65}} - \frac{6}{5\sqrt{65}} - \frac{8}{5\sqrt{65}} + \frac{8}{5\sqrt{65}} = 0 \]
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