Can a matrix of linear forms always be written as the sum of rank one matrices?

In summary, a matrix of linear forms is a rectangular array of numbers or symbols that represents a system of linear equations. These matrices are useful for solving linear equations in various fields of mathematics, engineering, and science. To multiply matrices of linear forms, the number of columns in the first matrix must match the number of rows in the second matrix. They cannot be used to represent non-linear equations, but can be used to approximate them. There are many real-world applications of matrices of linear forms, including in computer graphics, data analysis, and economics.
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Why is it a (for example) 3x3 matrix of linear forms cannot necessarily be written as the sum of at most 3 rank one matrices of linear forms but the statement is true if "linear forms" is replaced with scalars? Does it have something to do with the 2x2 minors being calculated differently when the entries are linear forms versus scalars? For example:

s t 0
0 s t
0 0 s

cannot be written as the sum of 3 rank one matrices of linear forms.
 
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Related to Can a matrix of linear forms always be written as the sum of rank one matrices?

1. What is a matrix of linear forms?

A matrix of linear forms is a rectangular array of numbers or symbols, arranged in rows and columns, that represents a system of linear equations. Each element in the matrix is a linear form, which is an expression of the form ax + by + cz + ... where a, b, c, etc. are coefficients and x, y, z, etc. are variables.

2. What is the purpose of using matrices of linear forms?

Matrices of linear forms are used to solve systems of linear equations, which are often encountered in various fields of mathematics, engineering, and science. By representing a system of equations in matrix form, it becomes easier to manipulate and solve using techniques such as Gaussian elimination and Cramer's rule.

3. How do you multiply matrices of linear forms?

To multiply two matrices of linear forms, the number of columns in the first matrix must match the number of rows in the second matrix. Then, for each entry in the resulting matrix, multiply the corresponding row in the first matrix by the corresponding column in the second matrix, and sum the products. The resulting number is the entry in the new matrix.

4. Can matrices of linear forms be used to represent non-linear equations?

No, matrices of linear forms can only represent systems of linear equations. Non-linear equations involve terms with higher powers, such as x^2 or sin(x), which cannot be expressed as linear forms. However, matrices of linear forms can be used to approximate non-linear equations by breaking them down into a series of linear equations.

5. Are there any real-world applications of matrices of linear forms?

Yes, matrices of linear forms have many practical applications, such as in computer graphics, data analysis, and optimization problems. They are also used in economics to model and solve systems of equations representing supply and demand, production costs, and other economic factors.

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