Page 102 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (b) each column of B which contains the leading non-zero entry of some row has all its
other entries 0.

5.5 Keywords

Equivalent: Two systems of linear equations are equivalent if each equation in each system is a
linear combination of the equations in the other system.
Row-equivalent: If A and B are m × n matrices over the field F, we say that B is row-equivalent
to A if B is obtained from A by a finite sequence of elementary row operations.

5.6 Review Questions

1. Find all solutions to the system of equations
(1 – i)x – ix = 0
1 2
2x + (1 – i)x = 0
1 2
3 1 2
2. Let A = 2 1 1
1 3 0

For which triples (y , y , y ) does this system AX = Y have a solution?
1 2 3
Answers: Self Assessment

0 1 1/4
1. Row-reduced matrix is 1 0 3/8 the solution is x = x = x = 0
1 2 3
0 0 3/8

64
0 1
24
5 67 5
3. Row-reduced matrix is 1 0 the solution is x 1 C ,x 2 C ,x 3 C where C is a
4 24 4
0 0 0
0 0 0
constant.
0 1
4. Row-reduced matrix is 0 0 . The solution is x = x = 0
1 2
1 0

5.7 Further Readings

Books Kenneth Hoffman and Ray Kunze Linear Algebra
I.N. Herstein Topics in Algebra

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