Linear dependence of vectors and row-reduced form matrices

[brad sue]

Hi ,
I have hard time understanding the concept of vector, matriecs in relation with rank and independent/dependent columns / rows.
I have two question that I would like to have help:
The set of vectors {v₁,v₂,v₃,v₄} would be linearly dependent if
1-v₃ is the zero vector
2- each of the vectors is in R³
3- the row reduced form of the matrix whose columns are the given vectors has rank less than 4
4-the row reduced form of the matrix whose rows are the given vectors has rank less than 4
5- any of the above would imply dependence
I would chose 3 or 4:
Please can you help me to define the expressions "the row reduced form of the matrix whose columns are the given vectors"
Is that means that the matrix would be like
[a b c d]
[e f g h]
[i j k l ]
where a,e,i are the coordinates of v₁
b,f,j are components of v₂
c,g,k are components of v₃
d,h,l are components of v₄?
Thank you very much
B.

 

[HallsofIvy]

Hi ,
I have hard time understanding the concept of vector, matriecs in relation with rank and independent/dependent columns / rows.
I have two question that I would like to have help:
The set of vectors {v₁,v₂,v₃,v₄} would be linearly dependent if
1-v₃ is the zero vector
2- each of the vectors is in R³
3- the row reduced form of the matrix whose columns are the given vectors has rank less than 4
4-the row reduced form of the matrix whose rows are the given vectors has rank less than 4
5- any of the above would imply dependence
I would chose 3 or 4:[\quote]
Be careful- you want dependence, not independence.
]quote]Please can you help me to define the expressions "the row reduced form of the matrix whose columns are the given vectors"
Is that means that the matrix would be like
[a b c d]
[e f g h]
[i j k l ]
where a,e,i are the coordinates of v₁
b,f,j are components of v₂
c,g,k are components of v₃
d,h,l are components of v₄?
Thank you very much
B.

That is the "matrix whose columns are the given vectors", now you would need to row-reduce it.

 

[brad sue]

That is the "matrix whose columns are the given vectors", now you would need to row-reduce it.

Thank you very much . But please , can you tell how do I need to reason in order to solve those kind of problem?

Thank you
B