Question 26364
Consider the accompanying matrix. Use the test for linear independence to find a basis for the space spanned by the rows of the matrix. Suppose that this matrix is augmented matrix for a system of equations. What is the rank of this systen? Which equations can be discarded? 
{[1 0 1 1
2 1 3 0
3 3 6 -3 
4 1 5 2]}
THE SECOND QUESTION FIRST ...AS PER THE SECOND QUESTION ,THIS IS AN AUGMENTED MATRIX.HENCE LAST COLUMN IS CONSTANTS COLUMN.
AND THE FIRST 3 COLUMNS ARE COEFFICIENT MATRIX.BUT THERE ARE 4  ROWS.THAT IS THERE ARE 3 UNKNOWNS AND 4 EQNS.LET US FIND RANK OF ASUGMENTED MATRIX...
		1	0	1	1				
		2	1	3	0				
		3	3	6	-3				
		4	1	5	2				

R1=R1…………		1	0	1	1				
R2=R2-2R1…..		0	1	1	-2				
R3=R3-3R1…..		0	3	3	-6				
R4=R4-4R1……		0	1	1	-2				

R1=R1…………		1	0	1	1				
R2=R2…….…..		0	1	1	-2				
R3=R3-3R2…..		0	0	0	0				
R4=R4-4R1……		0	0	0	0				
									
HENCE RANK = 2									
EQNS.2 AND 4 ARE LEADING TO SAME RESULT.									
AND EQN 3 AND 4 ARE ALL ZEROES . SO WE CAN DISCARD EQNS.3 AND 4 IN THIS SYSTEM. 									
SO WE REALLY HAVE 2 INDEPENDENT EQNS. IN 3 UNKNOWNS LEADING TO INFINITE SOLUTIONS.									
NOW COMING TO YOUR FIRST QUESTION ,THE DIMENSIONS OF THE BASIS IS NOT GIVEN.
TAKING 4 DIMENSIONAL BASIS FOR 4 EQNS.,WE GET 									
									
EQN.1 …..R1=		A+C+D							
EQN2…….R2=		2A+B+3C							
EQN.3……R3=		3A+3B+6C-3D						
EQN.4……R4=		4A+B+5C+2D							
									
BUT WE GOT R4-4R1=B+C-2D=R2-2R1……..OR……….R4=R2+2R1									
AND…………..R3-3R2+3R1=0………………….OR……….R3=3R2-3R1									
THAT IS TAKING R1 AND R2 AS 2 INDEPENDENT EQNS. WE SHOWED R3 AND R4 AS A LINEAR COMBINATION OF 									
R1 AND R2.	
HENCE THE BASIS FOR THESE SET OF 4 EQNS.CAN BE TAKEN AS 
R1=A+C+D....AND.....R2=2A+B+3C