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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/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/02%3A_Matrices/2.06%3A__The_Identity_and_InversesThere is a special matrix, denoted I , which is called to as the identity matrix
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/MAT_149%3A_Topics_in_Finite_Mathematics_(Holz)/03%3A_Systems_and_Matrices/3.05%3A_Matrix_Inverses/3.5.01%3A__The_Identity_and_InversesThere is a special matrix, denoted I , which is called to as the identity matrix
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/02%3A_Matrices/2.06%3A__The_Identity_and_InversesThere is a special matrix, denoted I , which is called to as the identity matrix
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.06%3A_Special_FunctionsThere are a great many functions that fail the horizontal line test which we nevertheless seem to have inverse functions for. For example, x^2 fails HLT but the square root of x is a pretty reasonabl...There are a great many functions that fail the horizontal line test which we nevertheless seem to have inverse functions for. For example, x^2 fails HLT but the square root of x is a pretty reasonable inverse for it – one just needs to be careful about the “plus or minus” issue. This apparent contradiction can be resolved using the notion of restriction.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/12%3A_Matrices_and_Determinants/12.06%3A__The_Identity_and_InversesThere is a special matrix, denoted I , which is called to as the identity matrix
- 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/02%3A_Matrices/2.06%3A__The_Identity_and_InversesThere is a special matrix, denoted I , which is called to as the identity matrix
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/18%3A_Orthonormal_Bases_and_Complements/18.01%3A_Properties_of_the_Standard_BasisThe standard notion of the length of a vector x=(x1,x2,…,xn) ∈ Rn is Given column vectors v and w, we have seen that the dot product v⋅w is the...The standard notion of the length of a vector x=(x1,x2,…,xn) ∈ Rn is Given column vectors v and w, we have seen that the dot product v⋅w is the same as the matrix multiplication vTw. In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. ΠiΠj=eieTiejeTj=eiδijeTj.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.04%3A_The_Dirac_Delta_FunctionThe Dirac delta function, δ(x) this is one example of what is known as a generalized function, or a distribution. Dirac had introduced this function in the 1930′s in his study of quantum mechanics a...The Dirac delta function, δ(x) this is one example of what is known as a generalized function, or a distribution. Dirac had introduced this function in the 1930′s in his study of quantum mechanics as a useful tool. It was later studied in a general theory of distributions and found to be more than a simple tool used by physicists. The Dirac delta function, as any distribution, only makes sense under an integral.
- 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.
