diff --git a/source/linear-algebra/source/01-LE/01.ptx b/source/linear-algebra/source/01-LE/01.ptx index 98ec945bd..c2ee9e260 100644 --- a/source/linear-algebra/source/01-LE/01.ptx +++ b/source/linear-algebra/source/01-LE/01.ptx @@ -58,70 +58,6 @@ Class Activities - - - -

A matrixmatrix is an m\times n array of real numbers -with m rows and n columns: - - \left[\begin{array}{cccc} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} \\ - \end{array}\right] - = -\left[\begin{array}{cccc} \vec v_1 & \vec v_2 & \cdots & \vec v_n\end{array}\right] -. -Frequently we will use matrices to describe an ordered list of -its column vectors: - - \left[\begin{array}{c} - a_{11} \\ - a_{21} \\ - \vdots \\ - a_{m1} \\ - \end{array}\right], - \left[\begin{array}{c} - a_{12} \\ - a_{22} \\ - \vdots \\ - a_{m2} \\ - \end{array}\right],\cdots, - \left[\begin{array}{c} - a_{1n} \\ - a_{2n} \\ - \vdots \\ - a_{mn} \\ - \end{array}\right] = - \vec v_1, \vec v_2, \cdots, \vec v_n -. -When order is irrelevant, we will use set notation: - - \left\{ - \left[\begin{array}{c} - a_{11} \\ - a_{21} \\ - \vdots \\ - a_{m1} \\ - \end{array}\right], - \left[\begin{array}{c} - a_{12} \\ - a_{22} \\ - \vdots \\ - a_{m2} \\ - \end{array}\right],\cdots, - \left[\begin{array}{c} - a_{1n} \\ - a_{2n} \\ - \vdots \\ - a_{mn} \\ - \end{array}\right]\right\} = - \{\vec v_1, \vec v_2, \cdots, \vec v_n\} -. -

- -

@@ -135,9 +71,12 @@ list of real numbers a_n \end{array}\right] . -We will find it useful to almost always typeset Euclidean vectors vertically, but the notation -\left[\begin{array}{cccc}a_1 & a_2 & \cdots & a_n\end{array}\right]^T is also -valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with +The notations (a_1, a_2, \cdots, a_n) and \langle a_1, a_2, \cdots, a_n) +are used in other mathematical contexts, but +we will find it useful to almost always typeset Euclidean vectors vertically. +However, the notation +\left[\begin{array}{cccc}a_1 & a_2 & \cdots & a_n\end{array}\right]^T +is also valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with n components is denoted as \mathbb R^n, and vectors are often described using the notation \vec v.

@@ -687,6 +626,69 @@ Then use these to describe the solution set + + +

A matrixmatrix is an m\times n array of real numbers +with m rows and n columns: + + \left[\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & a_{22} & \cdots & a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m1} & a_{m2} & \cdots & a_{mn} \\ + \end{array}\right] + = +\left[\begin{array}{cccc} \vec v_1 & \vec v_2 & \cdots & \vec v_n\end{array}\right] +. +Frequently we will use matrices to describe an ordered list of +column vectors: + + \left[\begin{array}{c} + a_{11} \\ + a_{21} \\ + \vdots \\ + a_{m1} \\ + \end{array}\right], + \left[\begin{array}{c} + a_{12} \\ + a_{22} \\ + \vdots \\ + a_{m2} \\ + \end{array}\right],\cdots, + \left[\begin{array}{c} + a_{1n} \\ + a_{2n} \\ + \vdots \\ + a_{mn} \\ + \end{array}\right] = + \vec v_1, \vec v_2, \cdots, \vec v_n +. +When order is irrelevant, we will use set notation: + + \left\{ + \left[\begin{array}{c} + a_{11} \\ + a_{21} \\ + \vdots \\ + a_{m1} \\ + \end{array}\right], + \left[\begin{array}{c} + a_{12} \\ + a_{22} \\ + \vdots \\ + a_{m2} \\ + \end{array}\right],\cdots, + \left[\begin{array}{c} + a_{1n} \\ + a_{2n} \\ + \vdots \\ + a_{mn} \\ + \end{array}\right]\right\} = + \{\vec v_1, \vec v_2, \cdots, \vec v_n\} +. +

+ +