Suppose A is a symmetric matrix of order n. Which of the following statements are true?
$\quad $ (1) Let k be an eigenvalue of A. Any basis $ S = \{u_1, ...u_k\} $ for the eigenspace $ E_k, $ must be orthogonal
$\quad $ (2) There is always a basis of $ \mathbb{R}^n $ consisting of eigenvectors of A.
$\quad $ (3) For every distinct eigenvalues $ k_1, k_2 $ of A, $ E_{k_{1}} $ is orthogonal to $ E_{k_{2}} $
$\quad $ (4) A can be orthogonally diagonalized, after applying Gram-Schmidt process to the chosen basis for each of the eigenspaces of A.
I can only prove that (3) and (4) is true. (4) is true because a symmetric matrix is a diagonalizable matrix, so each of the bases will be transformed to an orthonormal basis. (3) is true because the dot product of 2 eigenvectors of 2 distinct eigenvalues is zero.
How do I go about proving that (1),(2) are also true? Please correct me if my reasoning is incorrect. Thank you.
Hint:
For $1$, take $A=I_n$ and $k=1$ is its unique eigenvalue. We see that $S=(e_1+e_2,e_2, \ldots, e_n)$ is a basis for the eigenspace $E_1$ but isn't orthogonal.
For $2$, if $A \in \Bbb M_n(\Bbb R)$ and $A$ is symmetric, we know that $A$ is diagonalizable, so this is true (it is for every diagonalizable matrix on $\Bbb R$).
For $3$, by computing the dot product we see that it's true.
For $4$, since $A$ is diagonalizable and its eigenspaces are orthogonal, it's true.