Linear Algebra

Abstract Linear Algebra (Math416, UIUC)

• Lecture 2: Echelon Form and Reduced Echelon Form
• Study Notes: More on Elementary Matrix Operations and Elementary Matrices


• Lecture 3: Gaussian Elimination


• Lecture 4: Vector Spaces


• Lecture 5: Subspaces
• Book (1.3) Exercise 20: If \(W\) is a subspace of a vector space \(V\) and \(w_1,...,w_n \in W\), then \(a_1w_1 + a_2w_2 + ... + a_nw_n \in W\) for any scalars \(a_1, a_2,...,a_n\).


• Lecture 6: Span of a Subset
• Book (1.4) Theorem 1.5: The span of any subset \(S\) of a vector space \(V\) is a subspace of \(V\) that contains \(S\). Moreover, any subspace of \(V\) that contains \(S\) must also contain the span of \(S\).


• Lecture 7/8: Linear Dependance
• Book (1.5) Theorem 1.7: If \(S\) is a linearly independent subset of \(V\) and \(v\) is a vector in \(V\) but not \(S\). Then \(S \cup \{v\}\) is linearly dependent only if \(v \in span(S)\).
• Book (1.5) Exercise 21


• Lecture 9: Basis Vectors and the Replacement Theorem
• Book (1.6) Theorem 1.9: If \(V\) has a finite generating set, then \(V\) has a finite basis.
• Book (1.6) Theorem 1.7: Corollary 2
• Book (1.6) Exercise 11
• Book (1.6) Exercise 20


• Lecture 10: Linear Transformations


• Lecture 11/12: Null Space, Range, and Dimension Theorem
• Book (2.1) Theorem 2.2: If \(T\) is linear and \(\beta=\{v_1,...,v_n\}\) is basis for \(V\), then \(R(T) = span(T(\beta)) = span(\{T(v_1),...,T(v_n)\})\)
• External Exercise: Prove that if \(\dim(V) < \dim (W)\), then \(T\) cannot be onto.


• Lecture 12/13: Matrix Representation of Linear Transformations


• Lecture 14: Matrix Representation of Composition and Matrix Multiplication


• Lecture 15: Invertible Linear Transformations and Isomorphisms
• Book (2.4) Theorem 2.17 (Corollary): If \(T\) is invertible, then \(V\) is finite dimensional iff \(W\) is finite dimensional. In this case \(\dim V = \dim W\).


• Lecture 16: Inverse and Invertible Matrices


• Lecture 17: Change of Coordinates and Matrix Representations


• Lecture 18: Elementary Matrices
• Book (3.1-3.2) Study Notes


• Lecture 19/20: Determinants
• Lecture 21: Determinants and Row Operations
• Lecture 22: Determinants and Invertible Matrices
• Book (4.1-4.2) Study Notes
• Book (4.3) Theorem 4.8: For any \(A \in M_{n \times n}(\mathbf{F}\), \(\det(A^t) = \det(A)\).
• Book (4.3) Exercise 21


• Lecture 23: Eigenvalues and Diagonalizability
• Lecture 24: More Diagonalizability
• Lecture 25: Diagonalization and Eigenvalues
• Book (5.1) Exercise 15
• Book (5.2) Theorem 5.5


• Lecture 26/27: Markov Chains and Transition Matrices
• Book (5.3) Transition Matrices


• Lecture 28: Invariant and T-cyclic Subspaces


• Lecture 29/30: Inner Product Spaces and Norms
• Book (6.1) Theorem 6.1


• Lecture 31/32: Orthonormal Sets, Gram Schmidt and Vector Decomposition
• Book (6.2) Theorem 6.5
• Book (6.2) Exercise 11


• Lecture 33: Least Squares and Adjoint Maps
• Lecture 34/35: Normal and Self Adjoint Maps
• Book (6.3) Theorem 6.8
• Book (6.3) Theorem 6.9
• Book (6.3) Theorem 6.10
• Book (6.3) Theorem 6.11
• Book (6.3) Exercise 18


• Lecture 36: Isometries


• Lecture 37/38: The Jordan Canonical Form and Generalized Eigenvectors
• Lecture 39: Jordan Blocks and Generalized Eigenvectors
• Book (7.1) Study Notes

Essense of Linear Algebra (3Blue1Brown)

• Vectors
• Linear Combinations
• Linear Transformations
• Matrix Multiplication As Composition
• Determinants
• Inverse Matrices, Column Space and Null Space