System of linear equation in term of column vector

In summary: Your attachments are too small for me to read.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, arex_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
  • #1
DUET
55
0
Hello!

The following system of linear equations
3d924afa5682b1d557305e2ac1f37826.png

has been expressed in term of column vector in the following.
70027b64eee94f04f5b6ffdc37b29fe1.png
.
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.
 
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  • #2
DUET said:
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.
 
  • #3
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
[tex]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}[/tex]

Writing it as "rows" instead of "columns" would just be
[tex]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}[/tex]

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.)
 
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  • #5
HallsofIvy said:
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
[tex]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}[/tex]

Writing it as "rows" instead of "columns" would just be
[tex]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}[/tex]

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?
 
  • #6
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 [itex]f_i(v_j)= 1[/itex] 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.)
 
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1. What is a system of linear equations in terms of column vectors?

A system of linear equations in terms of column vectors is a set of equations where the variables are represented as column vectors. This means that each equation is written in the form of Ax = b, where A is a matrix of coefficients, x is a column vector of variables, and b is a column vector of constants.

2. How is a system of linear equations in terms of column vectors solved?

A system of linear equations in terms of column vectors can be solved using various methods, such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve manipulating the matrices and using algebraic operations to find the values of the variables.

3. What is the significance of using column vectors in a system of linear equations?

Using column vectors in a system of linear equations allows for a more efficient and organized way of solving the equations. It also allows for the use of matrix operations, which can simplify the process and make it easier to solve systems with a large number of equations and variables.

4. Can a system of linear equations in terms of column vectors have more than one solution?

Yes, a system of linear equations in terms of column vectors can have one, infinite, or no solutions. This depends on the nature of the equations and the values of the variables. For example, a system with fewer equations than variables will have infinite solutions, while a system with more equations than variables may have no solutions.

5. What is the connection between a system of linear equations in terms of column vectors and linear transformations?

A system of linear equations in terms of column vectors can be seen as a representation of a linear transformation. Each equation can be interpreted as a linear combination of the variables, and the solution to the system represents the point where all these linear combinations intersect. This connection helps in understanding and visualizing the solutions to the system of equations.

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