Linear Algebra and Its Applications
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Coordinate System and Change of Basis
Let \(\mathcal{B} = \{b_1, b_2, ..., b_n\}\) be a basis for a vector space \(V\). Then for \(x = [x_1, x_2, ..., x_n]^T\) in \(V\), there exists a unique set of scalars \(q_1, q_2, .
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Orthogonality and Projection
Orthogonality and Independence If \(\{u_1, u_2, ..., u_k\}\) are orthogonal to each other, then they are independent with each other. Orthonormality An \(m \times n\) matrix \(U\) has orthonormal columns if and only if \(U^TU = I\).
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Gram-Schmidt Orthogonalization
Gram-Schmidt Orthogonalization The Gram-Schmidt process is a simple algorithm for producing orthogonal basis for any nonzero subspace of \(\R^n\). Given a basis \(\{ \x_1, \dots, \x_p \}\) for a nonzero subspace \(W\) of \(\R^n\), define
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Least Squares
Suppose we are solving the \(Ax = b\) problem. \(b\) does not always lie in the column space of \(A\). However, we can try to find within \(A\)’s column space a vector \(\hat x\) such that \(A\hat x\) best approximates \(b\).