- #1

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My book doesn't have any examples of how to do this with matrices so I don't know how to approach this.

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- Thread starter Dustinsfl
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In summary, a linear operator in R<sup>n x n</sup> is a function that uses matrix multiplication to map a vector in R<sup>n</sup> to another vector in R<sup>n</sup>. It differs from a matrix, which is a fixed representation of a linear operator. The inverse of a linear operator can be calculated using the inverse of its corresponding matrix. The eigenvalues and eigenvectors of a linear operator represent the scaling factor and direction of vectors that do not change when the operator is applied. Linear operators have various applications in fields such as physics, engineering, and computer science.

- #1

- 2,281

- 5

My book doesn't have any examples of how to do this with matrices so I don't know how to approach this.

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- #2

Mentor

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1) L(A + B) = L(A) + L(B)

2) L(cA) = cL(A)

Here A and B are n x n matrices and c is a scalar.

A linear operator in R^{n x n} is a function that maps a vector in R^{n} to another vector in R^{n}, using a matrix multiplication operation.

A linear operator is a function, while a matrix is a representation of that function. A linear operator can be represented by different matrices depending on the chosen basis, while a matrix is a fixed representation.

The inverse of a linear operator in R^{n x n} can be calculated using the inverse of its corresponding matrix. If the matrix is invertible, then the inverse exists and is also a linear operator.

The eigenvalues and eigenvectors of a linear operator in R^{n x n} represent the scaling factor and direction, respectively, of the vectors that do not change direction when the linear operator is applied.

Linear operators in R^{n x n} have a wide range of applications in various fields such as physics, engineering, and computer science. They are used for solving systems of linear equations, transformations in 3D graphics, signal processing, and quantum mechanics, among others.

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