Consistency of A System of Linear Equations

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A system of linear equations is considered consistent if it can be transformed into triangular form without contradictions, indicating that a solution exists. Triangular form allows for straightforward back substitution to find solutions, confirming consistency. A diagonal matrix is a special case of a triangular matrix that guarantees consistency, as it directly leads to unique solutions. Not all triangular matrices can be diagonalized, particularly if they have repeated eigenvalues or lack sufficient eigenvectors. Understanding these concepts is crucial for determining the consistency of linear systems.
Bashyboy
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Hello everyone,

I was just solving a problem in which I had to determine the system of linear equations were consistent. Evidently, if a system of linear equations is capable of being put into triangular form, with no contradictions present, then it must consistent. My question is, why is that so, why does being in triangular form imply consistency?
 
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You should be able to work it out - what does it mean for the system of equations to be "consistent"?

If the system is represented by a triangular matrix, what is the form of the corresponding equations?
Would these be consistent?

Can you see that a diagonal matrix means consistency?
Are there any triangular matrixes that cannot be diagonalized?
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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