# HOMOGENEOUS LINEAR SYSTEM

May 29th, 2015
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-A system of linear equations has no solution, or exactly one solution, or infinitely many solutions. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

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A system of linear equations hasno solution, orexactly one solution, orinfinitely many solutionsIf the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.Each different choice of x3 determines a (different) solution of the system, and every solution of the system is determined by a choice of x3. whenever there are some free variables in system of equations, sol of equations never be uniqueand have infinitely many solutionA matrix with only one column is called a column vector, or simply a vectorThe vector whose entries are all zero is called the zero vector and is denoted by 0.Linear CombinationsGiven vectors v1; v2; : : : ; vp in Rn and given scalars c1; c2; : : : ; cp, the vector y defined y =c1v1+ _ _ _ +Cpvpthe domain of T is Rn when A has n columns and the codomain of T is Rm when each column of A has m entries. The range of T is the set of all linear combinations of the columns of A, because each image T .x/ is of the form Ax.A matrix that is not invertible is sometimes called a singular matrix, and an invertiblematrix is called a nonsingular matrix

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