MTH 2810 - Linear Algebra and Optimization

• Find the dot product of two vectors and interpret the results geometrically.
• Use the dot product to determine the angle between two vectors and vector projections.
• Represent and manipulate vectors, planes, and subspaces algebraically and geometrically.
• Solve geometric problems involving lines and planes in space.
• Evaluate multivariable functions and interpret level sets, contour plots, and gradients.
• Compute and interpret partial derivatives, gradients, and directional derivatives of functions of several variables.
• Apply the multivariable Chain Rule to functions of several variables.
• Identify and classify critical points using partial derivatives, gradients, and Hessians.
• Apply Lagrange multipliers in constrained optimization problems.
• Perform and interpret matrix operations, including inverses, transposes, and products.
• Use elementary row operations to row reduce a matrix to row echelon and reduced row echelon form and interpret the result in context.
• Represent and analyze systems of linear equations in matrix form.
• Analyze linear transformations using derivative matrices and Jacobians.
• Determine the column space, null space, and rank of matrices.
• Determine a basis and the dimension of a subspace.
• Compute eigenvalues and eigenvectors and employ them in selected applications with interpretation.
• Use Singular Value Decomposition (SVD) to analyze and reduce dimensionality in data-driven models.

• Vectors, vector operations, and geometry in 3-space
• Vector projections
• Subspaces, span, basis, and dimension
• Orthogonality and projections onto subspaces
• Multivariable functions, partial derivatives, and contour plots
• Gradients, local approximations, and gradient descent
• Constrained optimization via Lagrange multipliers
• Linear transformations, matrix algebra, and derivative matrices
• Linear systems, column and null spaces
• Eigenvalues, eigenvectors, and applications
• Singular Value Decomposition (SVD)
• Hessians and applications to local extrema and optimization