Question 34258: Problem: Is it possible for the columns of a 4x3 matrix to be linearly dependent? if so, give an example and demonstrate the dependence. If not, prove it.
I understand what linearly dependent means, but I don't know what kind of a matrix we should use to prove it without coming up with a clear example with numbers. Please help.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Problem: Is it possible for the columns of a 4x3 matrix to be linearly dependent?
YES ,WE CAN ALWAYS WRITE TO SUIT THE REQUIREMENT..FOR..EXAMPLE...LET THE MATRIX A BE EQUAL TO
1,2,3
2,4,6
3,6,9
4,8,12
THE COLUMNS ARE
C1...........1,2,3,4
C2...........2,4,6,8
C3...........3,6,9,12
WE FIND EACH ELEMENT OF C2=2*EACH CORRESPONDING ELEMENT OF C1
WE FIND EACH ELEMENT OF C3=3*EACH CORRESPONDING ELEMENT OF C1
SO THE COLUMNS ARE LINEARLY DEPENDENT....
ANOTHER WAY TO MAKE SUCH MATRIX IS SAY A IS.............
1,2,3
4,5,9
6,7,13
8,9,17
THE COLUMNS ARE
C1...1,4,6,8
C2...2,5,7,9
C3...3,9,13,17
WE FIND EACH ELEMENT OF C3=SUM OF EACH CORRESPONDING ELEMENT OF C1 AND C2
if so, give an example and demonstrate the dependence. If not, prove it.
I understand what linearly dependent means, but I don't know what kind of a matrix we should use to prove it without coming up with a clear example with numbers. Please help.
IN GENERAL JUST MAKE NUMBERS IN ONE COLUMN TO BE SUM OR DIFFERENCE OF ELEMENTS IN OTHER 2 COLUMNS...SAY..A..IS...
X,Y,X-Y
P,Q,P-Q
T,U,T-U
E,F,E-F...ETC...
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