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The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.
Example of solving a 3-by-3 homogeneous matrix equation
Definition of matrix equation
A matrix equation is equivalent to a linear system
Example of solving a 3-by-3 matrix equation

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Example of solving a 3-by-3 homogeneous matrix equation

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Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrixClose match
Example of solving a 3-by-3 matrix equationClose match
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrixClose match
A homogeneous system has a nontrivial solution if and only if it has a free variable.Close match
Matrix equationsClose match
Definition of homogeneous linear system of equationsClose match
The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.Close match
Definition of trivial solution to a homogeneous linear system of equationsClose match
The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.Close match
Using matrices to solve linear systemsClose match
Definition of matrix equationClose match
A matrix equation is equivalent to a linear systemClose match
Geometric picture of a 3-by-3 linear systemClose match
Example of row reducing a 3-by-3 matrixClose match
Definition of nontrivial solution to a homogeneous linear system of equationsClose match
A linear system is equivalent to a matrix equation.Close match
Example of using the echelon form to determine if a linear system is consistent.Close match
The solutions to a homogeneous system of linear equations is a vector space.Close match
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