Summary

Table of Contents:

1. Gaussian Elimination and Systems of Linear Equations (Lecture 1)

   • Key Concept: Efficient procedure to solve a system of linear equations (SLE) using row operations: replacement, scaling, and interchange.
   • Example: Solving 2x2 systems to find unique solutions.
   • Main Formula: (Ax = b) representation and augmented matrices.

2. Vector Spaces and Linear Combinations (Lecture 2)

   • Key Concept: Introduction to vector spaces, span, and the concept of linear combinations.
   • Example: Combining vectors with weights to form new vectors.
   • Main Formula: $(\vec{v}=c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\dots +c_n\overrightarrow{v_n})$.

3. Linear Independence and Bases (Lecture 3)

   • Key Concept: Criteria for linear independence and the definition of bases.
   • Example: Determining the independence of vectors and constructing bases for vector spaces.
   • Main Formula: A set of vectors ($\{v_1,v_2,\dots ,v_n\}$) is linearly independent if ($c_1{v}_1+c_2{v}_2+\dots +c_n{v}_n=0$) only when ($c_1=c_2=\dots =c_n=0$).

4. Matrix Algebra and Transformations (Lecture 4)

   • Key Concept: Matrix-vector multiplication as a transformation, properties of matrix operations.
   • Example: Reflections and rotations as linear transformations.
   • Main Formula: ($A\vec{x}=\vec{b}$), where (A) is a matrix and ($\vec{x}$), ($\vec{b}$) are vectors.

5. The Inverse of a Matrix (Lecture 5)

   • Key Concept: Conditions for a matrix to be invertible and the process of finding an inverse.
   • Example: Finding the inverse of a 2x2 matrix.
   • Main Formula: ($A{A}^{-1}=A^{-1}A=I$), where (I) is the identity matrix.

6. Determinants (Lecture 6)

   • Key Concept: The determinant as a scalar attribute of matrices, affecting the invertibility.
   • Example: Calculating determinants via cofactor expansion.
   • Main Formula: ($\text{det}(A)=a_{11}{C}_{11}+a_{12}{C}_{12}+\dots +a_{1n}{C}_{1n}$).

7. Vector Spaces Revisited (Lecture 7)

   • Key Concept: More on vector spaces, subspaces, and their properties.
   • Example: Identifying subspaces within (${\mathbb{R}}^n$).
   • Main Formula: Conditions for a subset (W) of a vector space (V) to be a subspace.

8. Basis and Dimension (Lecture 8)

   • Key Concept: Definition of basis and dimension of vector spaces.
   • Example: Finding the basis and dimension of various vector spaces.
   • Main Formula: If (V) is a vector space with a basis of (n) vectors, then ($\text{dim}(V)=n$).

9. Eigenvalues and Eigenvectors (Lecture 9)

   • Key Concept: Understanding eigenvalues and eigenvectors, and their importance in transformations.
   • Example: Finding eigenvalues and eigenvectors of a given matrix.
   • Main Formula: ($A\vec{v}=\lambda \vec{v}$), where ($\lambda$) is an eigenvalue and ($\vec{v}$) is an eigenvector.

10. Diagonalization (Lecture 10)

    • Key Concept: The process of diagonalizing a matrix and its applications.
    • Example: Diagonalizing a matrix given its eigenvalues and eigenvectors.
    • Main Formula: ($P^{-1}AP=D$), where (D) is a diagonal matrix.

Things to Remember for the Exam:

• Row Operations and their effects on the system of equations.
• Criteria for linear independence and how to identify a basis for a vector space.
• The process for finding the inverse of a matrix and its implications on the system’s solutions.
• Determinant calculation methods and their significance in matrix properties.
• The definition and properties of vector spaces, including subspaces.
• **The