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Eigenspaces are a fundamental concept when dealing with matrices and linear transformations. They consist of all eigenvectors corresponding to a given eigenvalue, along with the zero vector. This space thus represents all directions where the transformation induced by the matrix acts like mere scaling by the eigenvalue. If a matrix has an eigenvalue, the associated eigenspace is a vector space itself. For our matrix example, the eigenspace corresponding to eigenvalue \( \lambda_1 = 1 \) is denoted as \( E_1 \), and is determined by solving the eigenvector equation \( (A - \lambda_1 I)x = 0 \), where \( A \) is the matrix, \( \lambda_1 \) is the eigenvalue, and \( I \) is the identity matrix. The solutions, in this case, form the set of all possible linear combinations of the found eigenvectors for \( \lambda_1 \). This means they span the eigenspace \( E_1 \). Understanding eigenspaces is crucial as they can reveal important structural properties of transformations represented by the matrix.

An orthogonal basis for a vector space is a set of vectors that are mutually perpendicular to each other, simplifying many calculations, including projections and expansions of vectors. When constructing an orthogonal basis for an eigenspace, like \( E_1 \) with eigenvectors \( \begin{bmatrix} 2 & 1 & 0 \end{bmatrix} \) and \( \begin{bmatrix} -3 & 0 & 1 \end{bmatrix} \), our goal is to develop a set of vectors that maintains span while ensuring mutual orthogonality. This process allows property advantageous for decomposing vectors and simplifying the matrix-representation of transformations. Because orthogonal vectors essentially operate independently, calculations such as inner products and projections become trivial tasks, greatly aiding in solving complex linear algebra systems.

The Gram-Schmidt Procedure is a method used to transform a set of vectors into an orthogonal set while maintaining their span within the space. It is particularly useful in linear algebra for ensuring that a basis is orthogonal, which simplifies many calculations. The process involves iteratively subtracting the projections of a vector onto other vectors in your recursively forming orthogonal set.

Given a set of linearly independent vectors, such as \( \begin{bmatrix} 2 & 1 & 0 \end{bmatrix} \) and \( \begin{bmatrix} -3 & 0 & 1 \end{bmatrix} \), the first vector \( u_1 \) remains unchanged. For the second vector, \( u_2 \), subtract its projection onto \( u_1 \) to make it orthogonal to \( u_1 \). The projection is calculated as the dot product of the second vector with \( u_1 \), divided by the dot product of \( u_1 \) with itself, and then multiplied by \( u_1 \). This forms a new orthogonal vector \( u_2 \), if calculated correctly. This method provides an orthogonal basis that makes vector operations much more manageable. The result is a set of vectors ready for extensive computational applications while maintaining the structural properties of the original space.