Column Space Example [[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]

1. Find Column Space ...
[1-203-43281423723-1204-3] $[[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]$

Solution:

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
3 $3$	2 $2$	8 $8$	1 $1$	4 $4$
2 $2$	3 $3$	7 $7$	2 $2$	3 $3$
-1 $-1$	2 $2$	0 $0$	4 $4$	-3 $-3$

Now, reduce the matrix to row echelon form

R2←R2-3×R1 $R_2 larr R_2-3xx R_1$

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
0 $0$	8 $8$	8 $8$	-8 $-8$	16 $16$
2 $2$	3 $3$	7 $7$	2 $2$	3 $3$
-1 $-1$	2 $2$	0 $0$	4 $4$	-3 $-3$

R3←R3-2×R1 $R_3 larr R_3-2xx R_1$

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
0 $0$	8 $8$	8 $8$	-8 $-8$	16 $16$
0 $0$	7 $7$	7 $7$	-4 $-4$	11 $11$
-1 $-1$	2 $2$	0 $0$	4 $4$	-3 $-3$

R4←R4+R1 $R_4 larr R_4+ R_1$

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
0 $0$	8 $8$	8 $8$	-8 $-8$	16 $16$
0 $0$	7 $7$	7 $7$	-4 $-4$	11 $11$
0 $0$	0 $0$	0 $0$	7 $7$	-7 $-7$

R3←R3-78×R2 $R_3 larr R_3-7/8xx R_2$

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
0 $0$	8 $8$	8 $8$	-8 $-8$	16 $16$
0 $0$	0 $0$	0 $0$	3 $3$	-3 $-3$
0 $0$	0 $0$	0 $0$	7 $7$	-7 $-7$

R4←R4-73×R3 $R_4 larr R_4-7/3xx R_3$

1 $1$	-2 $-2$	0 $0$	3 $3$	-4 $-4$
0 $0$	8 $8$	8 $8$	-8 $-8$	16 $16$
0 $0$	0 $0$	0 $0$	3 $3$	-3 $-3$
0 $0$	0 $0$	0 $0$	0 $0$	0 $0$

The rank of a matrix is the number of non all-zeros rows

∴Rank=3 $:. Rank = 3$

Column Space :
The matrix has 3 pivots and Pivots are in the columns 1,2 and 4.
We know that these columns in the original matrix define the column space of the matrix.

∴ $:.$ The Column Space is

[132-1],[-2232],[3124] $[[1],[3],[2],[-1]],[[-2],[2],[3],[2]],[[3],[1],[2],[4]]$

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example [1-203-43281423723-1204-3][[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]

1. Example [1-203-43281423723-1204-3] $[[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]$