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# lec2-2 - Singular matrices does a matrix always have an...

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Singular matrices... does a matrix always have an inverse? No! Matrices need not have an inverse If a matrix does not have an inverse, it is not invertible , or it is called singular Consider our expression for M - 1 , M - 1 = 1 det M C T M is singular if det M = 0 Geometric interpretation: If we consider the rows (or columns) of M to be vectors (consider a 3 × 3 matrix, the determinant is the volume Ω of a polyhedron described by the vectors. Then Ω = det M = 0 corresponds to the case where the three vectors lie all in a plane. This is related to the question of linear dependence/independence that we will address later. Patrick K. Schelling Introduction to Theoretical Methods

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Rotation matrix in two dimensions Linear transformations are important applications of linear algebra Coordinate transformations are important, including rotations Start with a vector ~ r = x ˆ i + y ˆ j and rotate by angle φ After rotation, x x 0 and y y 0 , or ~ r ~ R We can find the rotation matrix R ( φ ) and the transformation x 0 y 0 = cos φ - sin φ sin φ cos φ x y What is the inverse of this matrix? Think of the simplest idea Patrick K. Schelling Introduction to Theoretical Methods
Inverse of rotation matrix... rotate back by angle - φ ! We could rotate back with R - 1 ( φ ) = R ( - φ ) R ( φ ) = cos φ - sin φ sin φ cos φ R - 1 ( φ ) = R ( - φ ) = cos φ sin φ - sin φ cos φ Try it! R - 1 ( φ ) R ( φ ) = cos φ sin φ - sin φ cos φ cos φ - sin φ sin φ cos φ = 1 0 0 1 Patrick K. Schelling Introduction to Theoretical Methods

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Multiple rotations, and another (equivalent) picture It is easy to show that R ( θ ) R ( φ ) = R ( θ + φ ) (see Problem 25, section 6) Moreover, R ( γ ) R ( θ ) R ( φ ) = R ( γ + θ + φ ), etc. In another picture, we can imagine rotating the axes by an angle φ , and leaving vector ~ r = x ˆ i + y ˆ j fixed In this picture, x 0 and y 0 define point in new coordinate system, and are given by x 0 y 0 = cos φ sin φ - sin φ cos φ x y Both of these pictures are often used... In one rotating the vector, and the other rotating the coordinate system Rotation is example of a linear operator describing a
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lec2-2 - Singular matrices does a matrix always have an...

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