Question 30580
I want to find about(three things)the set, W={A=(1,0,0),B=(0,1,0),C=(0,0,1),D=(-7,8,-9)}
Is this linearly dependent?
YES FOR REASON GIVEN BY YOU ..WE CAN SHOW AS FOLLOWS EASILY
WE NOTICE THAT 
D=-7A+8B-9C ...OR
WE ARE ABLE TO FIND 
PA+QB+RC+SD=0...(-7A+8B-9C-D=0),WHILE
P,Q,R,S ARE NOT ALL EQUAL TO ZERO.HENCE THEY ARE LINEARLY DEPENDENT.
 Does this spans R3?
YES
 Is this a basis for R3?
FOR THIS YOU HAVE TO REFER DEFINITION OF BASIS..IT STIPULATES..W A SUBSET OF VECTOR SPACE R3 IS A BASIS FOR R3 IF
1.W CONSISTS OF  LINEARLY INDEPENDENT VECTORS
2.W GENERATES R3 OR EACH VECTOR IN R3 IS A LINEAR COMBINATION OF FINITE NUMBER OF ELEMENTS OF W .
SO WE HAVE CONDITION 2.SATISFIED ..BUT NOT CONDITION 1..SO W IS NOT A BASIS. 
I think this set spansR3 because the first three vectors span R3,
VERY GOOD
 so adding the last one does not change it.
CORRECT
 Also, I think it is linearly dependent as there are nontrivial solutions. 
PERFECT AS WE SAW ABOVE 
But, I am not sure if it is not a vasis for R3. Could you give me advice???
YOU HAVE DONE EXTREMELY WELL...KEEP IT UP..YOU JUST NEED A LITTLE POLISHING...GOOD LUCK.