Solution sets of linear systems - Linear Equations with Matrices

Solution sets of linear systems

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Notes:
A system of linear equations is homogeneous if we can write the matrix equation in the form Ax=0Ax=0.

A system of linear equations is nonhomogeneous if we can write the matrix equation in the form Ax=bAx=b.

We can express solution sets of linear systems in parametric vector form.

Here are the types of solutions a homogeneous system can have in parametric vector form:
1. With 1 free variable: x=tvx=tv
2. With 2 free variables: x=su+tvx=su+tv
3. With n free variables: x=av1+bv2++nvnx=av_1+bv_2+\cdots+nv_n

Here are the types of solutions a nonhomogeneous system can have in a parametric vector form:
1. With 1 free variable: x=p+tvx=p+tv
2. With 2 free variables: x=p+su+tvx=p+su+tv
3. With n free variables: x=p+av1+bv2++nvnx=p+av_1+bv_2+\cdots+nv_n
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Solution sets of linear systems

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