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Consider the following homogeneous linear system: x1 + 2x2 + 7x3 − 9x4 + 31x5 =...

Consider the following homogeneous linear system: x1 + 2x2 + 7x3 − 9x4 + 31x5 = 0 2x1 + 4x2 + 7x3 − 11x4 + 34x5 = 0 3x1 + 6x2 + 5x3 − 11x4 + 29x5 = 0 [10p] a) Find the rank of the coefficient matrix. [5p] b) Use part (a) to determine the dimension of the solution space. [10p] c) Find a basis for the solution space.

Expert Solution

augmented matrix is

1	2	7	-9	31
2	4	7	-11	34
3	6	5	-11	29

convert into Reduced Row Eschelon Form...

Add (-2 * row1) to row2

1	2	7	-9	31
0	0	-7	7	-28
3	6	5	-11	29

Add (-3 * row1) to row3

1	2	7	-9	31
0	0	-7	7	-28
0	0	-16	16	-64

Divide row2 by -7

1	2	7	-9	31
0	0	1	-1	4
0	0	-16	16	-64

Add (16 * row2) to row3

1	2	7	-9	31
0	0	1	-1	4
0	0	0	0	0

Add (-7 * row2) to row1

1	2	0	-2	3
0	0	1	-1	4
0	0	0	0	0

reduced system is

there is 2 pivot entry at first and third column hence rank of matrix is 2

there is no pivot entry at second , fourth and fifth column

hence dimension of solution space is 3

.

find l solution

...............

..........free variable

..................

.........free variable

.

milcah answered 2 years ago

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