Question

B is the coefficient matrix = 1 4 1 2

0 1 3 -4

0 2 6 7

2 9 5 -7

The part of the question that confuses me is the part that asks if the columns of B span R3 or not because I am only sure of R4.

Answer 1

Question Details

Can every vector in R4 be written as a linear combination

of the columns of matrix B above? Do the columns of B span R3?

B is the coefficient matrix = 1 4 1 2

0 1 3 -4

0 2 6 7

2 9 5 -7

The part of the question that confuses me is the part that asks if the columns of B span R3 or not because I am only sure of R4.

AT THE OUT SET PLEASE NOTE THE FOLLOWING

BASIS

TO FORM A BASIS FOR RN , WE NEED

1.WE NEED A SET OF N AND ONLY N VECTORS

2.THEY SHALL BE L.I.

TO SPAN RN ...

1.WE NEED A SET OF N OR MORE VECTORS

2.ANY N VECTORS IN THE SET SHALL BE L.I.

SO NOW WE HAVE R4..THAT IS N=4

SO WE NEED AT LEAST 4 VECTORS TO SPAN R4..WE HAVE 4 VECTORS ...OK..

NOW WE HAVE TO CHECK WHETHER THEY ARE L.I.

B=
1	4	1	2
0	1	3	-4
0	2	6	7
2	9	5	-7
NR4=R4-2R1
1	4	1	2
0	1	3	-4
0	2	6	7
0	1	3	-11
NR1=R1-4R2…NR3=R3-2R2…NR4=R4-R2
1	0	-11	18
0	1	3	-4
0	0	0	15
0	0	0	-7
WE FIND COLUMNS 1,2,3 ARE L.D.
AS THEY HAVE ZEROS IN THE III AND IV COORDINATES.
HENCE THEY CAN NOT SPAN R4…………ANSWER
THUS WE HAVE ONLY 3 INDEPENDENT COLUMN VECTORS
SAY COLUMNS.1,2 AND 4
NOW LET US CHECK WHETHER THEY SPAN R3…
WE HAVE 4 VECTORS ..
THAT IS MORE THAN 3 NEEDED TO SPAN R3….OK
3 OF THEM ARE L.I. ….OK
SO THEY SPAN R3 …ANSWER