What does this notation mean? Eigenvectors and Eigenvalues

Question

I am trying to solve this question but I couldn't understand what does $A(v_1+v_2)$ mean.

Could anybody help me with this? Thanks...

Answer 1

As $v_1, v_2 \in \mathbb{R}^2$, $v_1 + v_2 \in \mathbb{R}^2$, so $A(v_1 + v_2)$ is just the product of the matrix $A$ and the vector $v_1 + v_2$.

Answer 2

because $v_i$ is eigenvector and $\lambda_i$ is eigenvalue that means that

$A*v_1=\lambda_1*v_1$

$A*v_2=\lambda_2*v_2$

also please pay attention that $A(v_1+v_2)=A*v_1+A*v_2$

could you continue?just insert values