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Yes, the two independent eigenvectors $(1,-1)$ and $(1,1)$ form an eigenbasis for this specific example. They span the entire 2D space, allowing any vector to be expressed as a linear combination of these eigenvectors.

Conditions: The matrix is the reflection about $y=-x$.; The eigenvectors are linearly independent.; The dimension of the space is 2.

Yes, the two independent eigenvectors $(1,-1)$ and $(1,1)$ form an eigenbasis for this specific example. They span the entire 2D space, allowing any vector to be expressed as a linear combination of these eigenvectors.

Conditions: The matrix is the reflection about $y=-x$.; The eigenvectors are linearly independent.; The dimension of the space is 2.