Need Explination (linear Indpendance)

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A and B are both Matricies,

Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can be said about the columns of A?

Answer: The columns of A are Linearly Dependant.

Question: Why?
 
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stuckie27 said:
Suppose the last column o AB is entirely zero but B itself has no column of zeros. What can be said about the columns of A?

Answer: The columns of A are Linearly Dependant.

Question: Why?

Are A and B contravariant vecotrs? is AB their inner (scalar) product? If so, I'm not sure how it can have columns, please clarify.
 
edit, A and B are each a different Matrix.
 
take a column in AB, what do the entries represent? are they in some way related to linear combinations of the columns of A? (yes they are, that isn't rhetorical) and a column of zeroes might mean that... fill in the blanks using the definition of linear (in)dependence.
 
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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