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[abc][k] [10] [a,-b]

[01] [-c,k]

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Q: Find a transformation matrix which aligns a given vector Vabc with the vector K along the positive z axis?
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What is an eigenvalue?

If a linear transformation acts on a vector and the result is only a change in the vector's magnitude, not direction, that vector is called an eigenvector of that particular linear transformation, and the magnitude that the vector is changed by is called an eigenvalue of that eigenvector.Formulaically, this statement is expressed as Av=kv, where A is the linear transformation, vis the eigenvector, and k is the eigenvalue. Keep in mind that A is usually a matrix and k is a scalar multiple that must exist in the field of which is over the vector space in question.


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Vector matrix has both size and direction. There are different types of matrix namely the scalar matrix, the symmetric matrix, the square matrix and the column matrix.


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The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.


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ShearingFor shear mapping (visually similar to slanting), there are two possibilities. For a shear parallel to the x axis has x' = x + ky and y' = y; the shear matrix, applied to column vectors, is: A shear parallel to the y axis has x' = xand y' = y + kx, which has matrix form:ReflectionTo reflect a vector about a line that goes through the origin, let be a vector in the direction of the line: To reflect a point through a plane ax + by + cz = 0 (which goes through the origin), one can use , where is the 3x3 identity matrix and is the three-dimensional unit vector for the surface normal of the plane. If the L2 norm of a,b, and c is unity, the transformation matrix can be expressed as:Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation; it is an affine transformation.


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The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.


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The eigen values of a matirx are the values L such that Ax = Lxwhere A is a matrix, x is a vector, and L is a constant.The vector x is known as the eigenvector.


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