Row- and column vectors

Summary

• A row vector is a matrix with only one row (dimension $1×n$ ). If $u$ is a row vector then the element at column $j$ is denoted by $u_j$ .
• A column vector is a matrix with only one column (dimension $n×1$ ). If $u$ is a column vector then the element at row $i$ is denoted by $u_i$ .
• A vector is an $n$ -tuple (not a matrix). There is a one-to-one correspondence between a row vector and a vector and between a column vector and a vector.
• Notation: The entire row $i$ of a matrix $A$ is denoted by $a_{i,:}$ or $a_{i,⋅}$ . $a_{i,:}$ is a row vector.
• Notation: The entire column $j$ of a matrix $A$ is denoted by $a_{:,j}$ or $a_{⋅,j}$ . $a_{:,j}$ is a column vector.