Linear algebra
Course Content
Vectors and spaces
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Vectors
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Vector intro for linear algebra
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Real coordinate spaces
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Adding vectors algebraically & graphically
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Multiplying a vector by a scalar
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Vector examples
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Unit vectors intro
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Parametric representations of lines
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Linear combinations and spans
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Linear combinations and span
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Linear dependence and independence
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Introduction to linear independence
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More on linear independence
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Span and linear independence example
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Subspaces and the basis for a subspace
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Linear subspaces
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Basis of a subspace
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Vector dot and cross products
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Vector dot product and vector length
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Proving vector dot product properties
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Proof of the Cauchy-Schwarz inequality
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Vector triangle inequality
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Defining the angle between vectors
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Defining a plane in R3 with a point and normal vector
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Cross product introduction
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Proof: Relationship between cross product and sin of angle
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Dot and cross product comparison/intuition
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Vector triple product expansion (very optional)
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Normal vector from plane equation
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Point distance to plane
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Distance between planes
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Matrices for solving systems by elimination
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Solving a system of 3 equations and 4 variables using matrix row-echelon form
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Solving linear systems with matrices
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Using matrix row-echelon form in order to show a linear system has no solutions
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Null space and column space
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Matrix vector products
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Introduction to the null space of a matrix
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Null space 2: Calculating the null space of a matrix
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Null space 3: Relation to linear independence
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Column space of a matrix
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Null space and column space basis
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Visualizing a column space as a plane in R3
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Proof: Any subspace basis has same number of elements
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Dimension of the null space or nullity
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Dimension of the column space or rank
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Showing relation between basis cols and pivot cols
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Showing that the candidate basis does span C(A)
Matrix transformations
Alternate coordinate systems (bases)
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