Linear Algebra Exam 3 Review Problems and Solutions

Links and resources
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⏱️TIMESTAMPS⏱️
(0:00) Types of problems
(0:40) Characteristic polynomial for a 2x2 matrix using trace and determinant
(3:34) Find an eigenvector for a given eigenvalue
(8:37) Linear dependence of a collection of trigonometric functions
(11:53) Find a coordinate vector relative to a new ordered basis based on a linear combination using an old basis.
(16:16) Find a coordinate vector by solving a system of linear equations (or using an inverse matrix).
(21:40) Solve a linear difference equation initial-value problem
(31:22) Find a basis for the kernel of a linear differential operator (this leads to a general solution of the ordinary differential equation).
(35:25) Is a set of polynomials linearly independent or linearly dependent? Use coordinate vectors with respect to the standard basis of P3 and then do row operations to reduced row echelon form to determine the answer.
(40:24) Find the change of coordinates matrix from one ordered basis to another. Also find a coordinate vector.
(46:11) Rank-Nullity Theorem (Find a basis for Col(A), Row(A), and Nul(A). Find the rank of A: Rank(A). Relate to the Rank-Nullity Theorem by also noting the nullity of A.
(56:50) Apply the Rank-Nullity Theorem
(1:00:20) Prove the kernel of a linear transformation is a subspace of the domain vector space.
(1:04:57) Prove the set of images of a linearly dependent set under a linear transformation is another linearly dependent set.
(1:09:27) Eigenvectors corresponding to distinct real eigenvalues of a matrix A are linearly independent. Classify the origin for dY/dt = AY as a sink, source, or saddle point.
(1:12:46) Subspaces are vector spaces
(1:13:16) Basis Theorem (also Invertible Matrix Theorem)
(1:14:10) Eigenvectors corresponding to distinct eigenvalues cannot be scalar multiples of each other.
(1:14:52) Rank-Nullity Theorem
(1:15:30) Linear difference equation equilibrium point classification based on magnitudes of eigenvalues
(1:16:41) Rank-Nullity Theorem
(1:17:42) Definition of isomorphic vector spaces

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[keldonchase]

Hi Bill, thank you for all the videos you have provided.
They have been an immense help thus far; I really appreciate your efforts!!

I had a question regarding Problem 11 at 56:52:
In the video, you conclude that since the rank of the linear transformation T is lower than the number of rows, the linear transformation is not able to reach all of the vectors in R7 and thus is not onto.
If the rank were equal to or greater than the number of rows, would that confirm that the linear transformation is onto and therefore give the scientist certainty that the new homogeneous system will have a solution?