/sci/ - Science & Math

Say [math]A[/math] is a real nxn matrix that has two eigenvalues [math]\lambda_1[/math] and [math]\lambda_2[/math] and two eigenvectors associated to them respectively [math]X_1[/math] and [math]X_2[/math], and [math]c \neq 0[/math]. Which of the following is true:

1. If [math]\lambda_1 = \lambda_1[/math] then [math]cX_1 + X_2[/math] is an eigenvector of [math]A[/math];
2. If [math]\lambda_1 \neq \lambda_1[/math] then [math]cX_1 + X_2[/math] is an eigenvector of [math]A[/math];
3. If [math]\lambda_1 = c\lambda_1[/math] then [math]X_1[/math] and [math]X_2[/math] are linearly independent;
4. If [math]X_1[/math] and [math]X_2[/math] are linearly independent, then [math]\lambda_1 \neq \lambda_2[/math].

2 and 3 seem dumb, 4 is the converse of a known theorem (and it doesn't work for say a 3d shear). And 1 I can only find the converse. Pls help

>>12253229
This works, thanks
imma give you a small kiss for your help