Linear Algebra
Linear Algebra (Math416)
- 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 \(R(T) = span(T(\beta)) = span(\{T(v_1),...,T(v_n)\})\)
- 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
- Lecture 19/20: Determinants
- Lecture 21: Determinants and Row Operations
- Lecture 22: Determinants and Invertible Matrices
- Lecture 23: Eigenvalues and Diagonalizability
- Lecture 24: More Diagonalizability
- Lecture 25: Diagonalization and Eigenvalues
Essense of Linear Algebra (3Blue1Brown)
- Vectors
- Linear Combinations
- Linear Transformations
- Matrix Multiplication As Composition
- Determinants
- Inverse Matrices, Column Space and Null Space