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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.11%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.04%3A_Rotation_Matrices_and_Orthogonal_MatricesTrigonometry and the addition formula for cosine and sine results in \[\begin{aligned} x'&=r\cos(\theta+\psi) \\ &=r(\cos\theta\cos\psi -\sin\theta\sin\psi )\\&=x\cos\theta-y\sin\theta \\ y'&=r\sin(\t...Trigonometry and the addition formula for cosine and sine results in \[\begin{aligned} x'&=r\cos(\theta+\psi) \\ &=r(\cos\theta\cos\psi -\sin\theta\sin\psi )\\&=x\cos\theta-y\sin\theta \\ y'&=r\sin(\theta+\psi)\\&=r(\sin\theta\cos\psi+\cos\theta\sin\psi) \\ &=x\sin\theta+y\cos\theta.\end{aligned} \nonumber \] Writing the equations for \(x'\) and \(y'\) in matrix form, we have \[\left(\begin{array}{c}x'\\y'\end{array}\right)=\left(\begin{array}{rr}\cos\theta&-\sin\theta \\ \sin\theta&\cos\theta\…
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/04%3A_R/4.11%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/04%3A_R/4.11%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/04%3A_R/4.08%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/04%3A_R/4.11%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
