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Daaavde
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Is it correct to state that a diagonalizable endomorphism has always kernel = {0}?
A diagonalizable endomorphism is a linear transformation on a vector space that can be represented by a diagonal matrix with respect to some basis. This means that the transformation only stretches or shrinks each basis vector by a certain scalar value.
A null space, also known as a kernel, is the set of all vectors that are mapped to the zero vector by a given linear transformation. In other words, it is the set of all inputs that result in an output of zero.
Having a trivial null space means that the only vector that is mapped to the zero vector is the zero vector itself. This is important because it ensures that the linear transformation is one-to-one, meaning that each input has a unique output. This is necessary for certain applications, such as solving systems of linear equations.
An endomorphism is diagonalizable if and only if it has a full set of linearly independent eigenvectors. This means that the eigenvectors span the entire vector space and can be used as a basis for the transformation's matrix representation.
No, a non-square matrix cannot be diagonalizable. In order for a matrix to be diagonalizable, it must have the same number of rows and columns, meaning it must be a square matrix. This is necessary for the matrix to have a full set of linearly independent eigenvectors.