Linear Algebra Exam 1 Review Problems and Solutions

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⏱️TIMESTAMPS⏱️
(0:00) Introduction
(0:40) Vector algebra, dot product, norm, angle
(9:55) Scalar difference equation initial value problem
(13:21) Define matrix/vector multiplication Ax
(19:41) Solve a linear system of 2 equations and 2 unknowns and geometrically interpret the answer as the intersection of two points
(28:24) Change of coordinates: convert a linear equation to the new coordinates
(31:57) Interpret linear system as a matrix equation
(33:13) Find the kernel of a linear transformation from R^3 to R^3 using elementary row operations to reduced row echelon form (write the subspace in parametric vector form as linear combinations of vectors)
(42:35) T is not one to one
(43:47) Find the image of the linear transformation as the span of a set of vectors (write the subspace in parametric vector form as a linear combination of vectors)
(47:20) Find a formula for a reflection composed with a rotation
(54:14) Matrix multiplication to confirm the answers
(56:10) Solve a system of 3 equations and 4 unknowns by using a TI graphing calculator to put an augmented matrix in reduced row echelon form (it turns out there are 2 free variables and 2 basic variables). The solution set represents a 2-dimensional plane in 4-dimensional space.

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