Linear Algebra Please show work Thank you Let lambda be an e

Linear Algebra. Please show work. Thank you.

Let lambda be an eigenvalue of the matrix A M_n(R). If x_1, x_2, and x_3 are eigenvectors associated with lambda, prove that y = c_1x_1+c_2x_2+c_3x_3 is also an eigenvector of A associated with lambda.

Solution

We can represent any vector x R _n as a linear combination of the eigenvectors of A if and only if A has 3 eigenvectors that form a linearly independent set vectors . In the words, if the previous is true, and letting x₁, x₂, x₃ denote 3 linearly independent eigenvectors, then for any given x R n we can find some complex numbers c₁, c₂, c₃ such that x = c_1x1 + c_2x2 + c₃x₃.