The image presents a problem with the following details:

• Given: $A^2 = A$
• Questions:
  1. Which of the four subspaces contains eigenvectors with $\lambda = 1$?
  2. Which subspace contains eigenvectors with $\lambda = 0$?
  3. From the dimensions of these subspaces, the matrix $A$ has a full set of independent eigenvectors. Thus, a matrix with $A^2 = A$ can be diagonalized.

Solution:

1. Eigenvectors with $\lambda = 1$

• Projection Interpretation: The equation $A^2 = A$ implies that $A$ is a projection matrix. Projection matrices have eigenvalues either 0 or 1.
• Subspace for $\lambda = 1$: The eigenvectors corresponding to $\lambda = 1$ lie in the column space (also called the image or range) of $A$. These are the vectors that are mapped to themselves by $A$ (i.e., $Av = v$).

2. Eigenvectors with $\lambda = 0$

• Subspace for $\lambda = 0$: The eigenvectors corresponding to $\lambda = 0$ lie in the null space (or kernel) of $A$. These are the vectors that are mapped to the zero vector by $A$ (i.e., $Av = 0$).

3. Diagonalizability of $A$

• Dimensionality Consideration: For a matrix to be diagonalizable, it must have a full set of linearly independent eigenvectors. Given that $A$ has eigenvalues 0 and 1 and is a projection matrix, the sum of the dimensions of the null space and column space should equal the dimension of the entire space. This means that $A$ has as many independent eigenvectors as its dimension, and thus $A$ is diagonalizable.

Final Summary:

• Eigenvectors with $\lambda = 1$ are in the column space of $A$.
• Eigenvectors with $\lambda = 0$ are in the null space of $A$.
• Diagonalizability: Since $A$ is a projection matrix with $A^2 = A$, it is diagonalizable.

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Would you like further explanation or have any other questions?

Related Questions:

1. How can you verify if a given matrix is a projection matrix?
2. What properties do projection matrices have regarding their eigenvalues?
3. How can the diagonalization of a projection matrix be interpreted geometrically?
4. What is the significance of the null space and column space in the context of linear transformations?
5. How do the rank and nullity of a matrix relate to its diagonalization?

Tip:

To check if a matrix is diagonalizable, ensure that the sum of the dimensions of the eigenspaces matches the size of the matrix. This guarantees a complete set of linearly independent eigenvectors.