Vector Placing in Transformation Matrix: Rows vs. Columns Explained

[transgalactic]

in previous question
i was tald that in the process of building the transformation matrix

i should have put the vectors of the new base as columns and not as rows

in what cases and in what formes i put them as rows??
in what cases and in what formes i put them as columns??

 

[Marco_84]

in previous question
i was tald that in the process of building the transformation matrix

i should have put the vectors of the new base as columns and not as rows

in what cases and in what formes i put them as rows??
in what cases and in what formes i put them as columns??

Think about transposition...


regards

marco

 

[CompuChip]

Also, you can always check by applying the transformation matrix to one of your basis elements. If you want A to transform the vector v into the vector w, you can check that indeed
Av = w
and/or
A^-1 w = v.

For example, if you write down a general matrix
$$A = \begin{pmatrix} a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots & \ddots & \cdots \end{pmatrix}$$
you can apply it to the (old) basis vector
$$e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \vdots \end{pmatrix}$$
and just do the multiplication explicitly. You will get a result expressed in the $a_{ij}$ which will in fact be just a complete row or column from A. On the other hand, it should also be the new basis vector. So you can read off whether you need to put it in a column, or in a row.

 

[transgalactic]

can you give an actual example to this
and in simpler words

because i can't understand how this multiplication effects the desition to put the vectors by rows or by columns

using your multiplication i would get the first row of this matrix
now what??

 

[HallsofIvy]

He said: "Try both and see which gives what you want!"