System of linear equation in term of column vector

[DUET]

Hello!

The following system of linear equations

has been expressed in term of column vector in the following.

.
How can I express the system of linear equations in term of row vector?


In addition, What is the field of scalars? I would request to explain it.

Thanks in advance.

 

[Mark44]

Hello!

The following system of linear equations
View attachment 61063
has been expressed in term of column vector in the following.
View attachment 61064.
How can I express the system of linear equations in term of row vector?


In addition, What is the field of scalars? I would request to explain it.

Thanks in advance.

Your attachments are too small for me to read.

 

[HallsofIvy]

Writing that same system of equations in terms of "row vectors" only means that your write each "column" as a row. For example, your equations, in terms of "column" vectors, are
$$x_1\begin{bmatrix}a_{11} \\ a_{21} \\ \cdot\\ \cdot\\ \cdot \\ a_{m1} \end{bmatrix}+ x_2\begin{bmatrix}a_{12} \\ a_{22} \\ \cdot\\ \cdot\\ \cdot \\ a_{m2}\end{bmatrix}+ \cdot\cdot\cdot+ x_n\begin{bmatrix}a_{1n} \\ a_{2n} \\ \cdot\\ \cdot\\ \cdot \\ a_{mn} \end{bmatrix}= \begin{bmatrix}b_1 \\ b_2 \\ \cdot\\ \cdot\\ \cdot \\ b_m \end{bmatrix}$$

Writing it as "rows" instead of "columns" would just be
$$x_1\begin{bmatrix}a_{11} & a_{21} & \cdot\cdot\cdot a_{m1} \end{bmatrix}+ x_2\begin{bmatrix}a_{12} & a_{22} & \cdot & \cdot & \cdot a_{m2}\end{bmatrix}+ \cdot\cdot\cdot+ x_n\begin{bmatrix}a_{1n} & a_{2n} & \cdot &\cdot & \cdot a_{mn} \end{bmatrix}= \begin{bmatrix}b_1 & b_2 & \cdot & \cdot & \cdot b_m \end{bmatrix}$$

Unless something is said to the contrary, the "field of scalars" is assumed to be the field of real numbers.

(The field of complex numbers is sometimes used and, less often, the field of rational numbers. But those should be given explicitely.)

 

[MinnesotaState]

https://www.physicsforums.com/attachment.php?attachmentid=61063&d=1377183233

https://www.physicsforums.com/attachment.php?attachmentid=61064&d=1377183233

 

[DUET]

Writing that same system of equations in terms of "row vectors" only means that your write each "column" as a row. For example, your equations, in terms of "column" vectors, are
$$x_1\begin{bmatrix}a_{11} \\ a_{21} \\ \cdot\\ \cdot\\ \cdot \\ a_{m1} \end{bmatrix}+ x_2\begin{bmatrix}a_{12} \\ a_{22} \\ \cdot\\ \cdot\\ \cdot \\ a_{m2}\end{bmatrix}+ \cdot\cdot\cdot+ x_n\begin{bmatrix}a_{1n} \\ a_{2n} \\ \cdot\\ \cdot\\ \cdot \\ a_{mn} \end{bmatrix}= \begin{bmatrix}b_1 \\ b_2 \\ \cdot\\ \cdot\\ \cdot \\ b_m \end{bmatrix}$$

Writing it as "rows" instead of "columns" would just be
$$x_1\begin{bmatrix}a_{11} & a_{21} & \cdot\cdot\cdot a_{m1} \end{bmatrix}+ x_2\begin{bmatrix}a_{12} & a_{22} & \cdot & \cdot & \cdot a_{m2}\end{bmatrix}+ \cdot\cdot\cdot+ x_n\begin{bmatrix}a_{1n} & a_{2n} & \cdot &\cdot & \cdot a_{mn} \end{bmatrix}= \begin{bmatrix}b_1 & b_2 & \cdot & \cdot & \cdot b_m \end{bmatrix}$$

Unless something is said to the contrary, the "field of scalars" is assumed to be the field of real numbers.

(The field of complex numbers is sometimes used and, less often, the field of rational numbers. But those should be given explicitely.)

Is there any difference between the two expression. If there is no difference then which is more convenient and why?

 

[HallsofIvy]

It is entirely a matter of which is easier to read and/or write. Generally, book printers find it easier to write in horizontal lines (in language in which words and sentences are written horizontally!).

(Some authors use "horizontal" and "vertical" placement to distinguish between "vectors" and "co-vectors". Given any vector space, its "dual" is the set of all linear functionals on it- linear functions that map each vector to a number. If V is an n dimensional vector space, its "dual", V*, is also an n dimensional vector space with "sum" defined as (f+ g)(v)= f(v)+ g(v) and "scalar multiplication" by (af)(v)= a(f(v)). It can be shown that, given a basis for V, there is a "natural basis" for V* defined by $f_i(v_j)= 1$ if i= j, 0 other wise. Of course, using the basis for V, we can write any vector as n-numbers. and using that basis for V* we can write any "co-vector" (function in the dual of V) as n-numbers. If we agree to write vectors in V "vertically" and co-vectors in V* horizontally, then we can write f(v) as the matrix product of the "row matrix" representing f and the "column matrix" representing v.)