Given a span of V, you can find this by finding the kernel of the span turned on its side (i.e. the vector that is perpendicular to both spanning vectors)! Angle between Two Vectors Cauchy-Schwarz Inequality ensures that this value is defined. Cauchy-Schwarz Inequality Gram-Schmidt Process- an algorithm for producing an orthonormal basis from any basis. For the first vector, simply divide it by its length to create a unit vector. To find the next basis vector, first find This becomes in the general case. Then, This procedure is simply repeated for every vector in the original basis. Keep in mind that, to simply the calculation, any can multiplied by a scalar (becomes a unit vector, so won’t effect the end result just the difficulty of the calculation). Orthogonal Transformations and Orthogonal Matrices Orthogonal Transformation- A transformation, T, from Rn to Rn that preserves the length of vectors. Orthogonal transformations preserve orthogonality and angles in general (Pythagorean theorem proof). Useful Relations If T:RnRn is orthogonal and , then Orthogonal Matrices- The transformation matrix of an orthogonal transformation. Properties: The product, AB, of two orthogonal n x n matrices A and B is orthogonal The inverse A-1 of an orthogonal n x n matrix A is orthogonal A matrix is orthogonal ATA=In or, equivalently, A-1=AT. The columns of an orthogonal matrix form an orthonormal basis of Rn. The Transpose of a Matrix The matrix created by taking the columns of the original matrix and making them the rows of a new matrix (and therefore the rows become columns). More officially, the transpose AT of A is the n x m matrix whose ijth entry is the jith entry of A. Symmetric- A square matrix A AT=A Skew Symmetric- A square matrix A AT= -A
Given a span of V, you can find this by finding the kernel of the span turned on its side (i.e. the vector that is perpendicular to both spanning vectors)! Angle between Two Vectors Cauchy-Schwarz Inequality ensures that this value is defined. Cauchy-Schwarz Inequality Gram-Schmidt Process- an algorithm for producing an orthonormal basis from any basis. For the first vector, simply divide it by its length to create a unit vector. To find the next basis vector, first find This becomes in the general case. Then, This procedure is simply repeated for every vector in the original basis. Keep in mind that, to simply the calculation, any can multiplied by a scalar (becomes a unit vector, so won’t effect the end result just the difficulty of the calculation). Orthogonal Transformations and Orthogonal Matrices Orthogonal Transformation- A transformation, T, from Rn to Rn that preserves the length of vectors. Orthogonal transformations preserve orthogonality and angles in general (Pythagorean theorem proof). Useful Relations If T:RnRn is orthogonal and , then Orthogonal Matrices- The transformation matrix of an orthogonal transformation. Properties: The product, AB, of two orthogonal n x n matrices A and B is orthogonal The inverse A-1 of an orthogonal n x n matrix A is orthogonal A matrix is orthogonal ATA=In or, equivalently, A-1=AT. The columns of an orthogonal matrix form an orthonormal basis of Rn. The Transpose of a Matrix The matrix created by taking the columns of the original matrix and making them the rows of a new matrix (and therefore the rows become columns). More officially, the transpose AT of A is the n x m matrix whose ijth entry is the jith entry of A. Symmetric- A square matrix A AT=A Skew Symmetric- A square matrix A AT= -A