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3-space angle apup augmented matrix axis Chapter coefficient matrix components Compute coordinate system corresponding cross product defined definition denote determinant diagonal entries diagonal matrix dimension dot product echelon form eigenvalues eigenvectors eigenvectors belonging equal equivalent Example 1.2 Find all vectors formula function from Rn g o f geometric graph gth column Hence homogeneous system identity matrix implies independent set intersection invertible kernel length linear combination linear equations linear function linear mapping manipulations mapping from Rn n-tuple origin orthogonally diagonalizable polynomial position vectors principal axis theorem proof prove pth row quadratic form rank real number Rn to Rm rotation row space satisfying scalar multiples Section set of vectors Show Similarly space spanned spanning set subspace of Rn symmetric system of equations theorem tion u x v u,+i vector perpendicular vector space vectors in Rn x-axis z-axis