1
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage{amsmath, amssymb, amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{fancyhdr}
\usepackage{tcolorbox}
\usepackage{bm}
\setlength\parindent{0pt}
\pagestyle{fancy}
\fancyhead{}
\fancyfoot{}
\fancyhead[L]{\MakeUppercase{Linear Algebra (MAT240/247)}}
\fancyhead[R]{Ali Manasiya}
\fancyfoot[C]{\thepage}
\tcbuselibrary{most}
\renewcommand{\qedsymbol}{$\blacksquare$}

\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\C{\mathbb{C}}
\def\F{\mathbb{F}}
\def\P{\mathcal{P}}
\def\L{\mathcal{L}}
\def\M{\mathcal{M}}

I need the following but I don't know how to format this. (note there are two images) For the bottom image I have the following:

\[\sum_{k = 1}^n A_{1,k}x_k = 0\\
\vdots \\
\sum_{k = 1}^n A_{m,k}x_k = 0\]

But for the one on the top i got no clue. any help would be greatly appreciated. Thank you for your time

Image 1 enter image description here

2

enter image description here

Used are mathtools and nicematrix package:

\documentclass{article}
\usepackage{mathtools}
\usepackage{nicematrix}

\begin{document}

    \[
\mathcal{M}(T) = 
    \begin{pNiceMatrix}[first-row,first-col]
        & v_1 & \dotsm & v_k     & \dotsm & v_n \\
w_1     &     &        & A_{1,k} &        &     \\
\vdots  &     &        & \vdots  &        &     \\
w_1     &     &        & A_{m,k} &        &     \\
    \end{pNiceMatrix}.
    \]  

\begin{align*}
\sum_{k = 1}^n A_{1,k}x_k   & = 0               \\
                            & \vdotswithin{=}   \\
\sum_{k = 1}^n A_{m,k}x_k   & = 0
\end{align*}
\end{document}

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