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Dot Product, Norm, Orthogonality, Cross Product, Equations of Planes in R^3, Kernel & Image of a LT
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(a.k.a. Differential Equations with Linear Algebra, Lecture 9B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 9B. #linearalgebra #crossproduct #orthogonal).
(0:00) This is a lecture of miscellaneous but important topics in linear algebra
(1:08) Dot product of two vectors in n-dimensional space
(3:46) Properties of dot product
(4:09) Commutative property of dot product
(4:44) Distributive property of dot product over vector addition
(5:28) Associative property of dot product
(6:15) Non-negativity of the dot product of a vector with itself
(7:41) Norm of a vector in n-dimensional space (a.k.a. magnitude or length)
(9:04) Properties of norm (along with dot product). First: non-negativity
(9:32) Factoring out scalars (with an absolute value)
(9:54) Triangle inequality and visualization
(11:20) Cauchy-Schwartz Inequality
(12:20) Defining an angle between two vectors in n-dimensional space
(16:12) v and w are perpendicular if and only if their dot product is zero
(17:16) Definition of orthogonal vectors
(18:16) Equations of planes in 3-dimensional space
(19:14) Relationship to dot product
(20:26) Parallel planes
(21:04) Given a normal vector and a point in the plane, find an equation of the plane
(23:05) Cross product of two vectors in R^3
(24:50) Properties of cross product
(25:03) Cross a vector with itself gives the zero vector
(25:24) Anticommutativity
(25:44) Distributive property over vector addition
(26:03) Magnitude of a cross product (relationship to the angle)
(26:58) Orthogonality (v x w is orthogonal to both v and w)
(27:41) Equation of a plane given 3 noncollinear points
(29:58) Linear transformations from R^3 to R
(31:39) It is “usually” onto so that the image of T is R: Im(T) = R, which is the column space of its matrix A
(33:19) It is never one-to-one (the kernel of T, which is the null space of its matrix A, will usually be a plane through the origin)
(34:33) Example where A = (3 2 -4)
(35:59) Graph the kernel as a plane in R^3
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