# 7.4 A Gallery of Solution Curves of Linear Systems

In the preceding section we saw that the eigenvalues and eigenvectors of the $n\times n$ matrix **A** are of central importance to the solutions of the homogeneous linear constant-coefficient system

Indeed, according to Theorem 1 from Section 7.3, if $\lambda $ is an eigenvalue of **A** and **v** is an eigenvector of **A** associated with $\lambda ,$ then

is a nontrivial solution of the system (1). Moreover, if **A** has `n` linearly independent eigenvectors ${\mathbf{\text{v}}}_{1},\text{}{\mathbf{\text{v}}}_{2},\text{}\dots ,$ ${\mathbf{\text{v}}}_{n}$ associated with its `n` eigenvalues ${\lambda}_{1},\text{}{\lambda}_{2},\text{}\dots ,$ ${\lambda}_{n},$ then in fact all solutions of the system (1) are given by linear combinations

where ${c}_{1},{c}_{2},\dots ,{c}_{n}$ are arbitrary constants. If the eigenvalues include complex conjugate ...

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