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lecture_13
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Table of Contents
The transpose of a matrix
We defined this in tutorial sheet 4:
The transpose of an $n\times m$ matrix $A$ is the $m\times n$ matrix $A^T$ whose $(i,j)$ entry is the $(j,i)$ entry of $A$. In other words, to get $A^T$ from $A$, you write the rows of $A$ as columns, and vice versa; equivalently, you reflect $A$ in its main diagonal.
For example, $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}\mat{a&b\\c&d}^T=\mat{a&c\\b&d}$ and $\mat{1&2&3\\4&5&6}^T=\mat{1&4\\2&5\\3&6}$.
Exercise: simple properties of the transpose
Prove that for any matrix $A$:
- $(A^T)^T=A$; and
- $(A+B)^T=A^T+B^T$ if $A$ and $B$ are matrices of the same size; and
- $(cA)^T=c(A^T)$ for any scalar $c$.
In tutorial sheet 4, we proved:
Lemma: transposes and row-column multiplication
If $a$ is a $1\times m$ row vector and $b$ is an $m\times 1$ column vector, then \[ ab=b^Ta^T.\]
lecture_13.1457347750.txt.gz · Last modified: by rupert