Linear Algebra for Team-Based Inquiry Learning: Legacy Edition

Learning Outcomes

• E: How can we solve systems of linear equations?

  • E1: I can translate back and forth between a system of linear equations, a vector equation, and the corresponding augmented matrix.

  • E2: I can explain why a matrix isn’t in reduced row echelon form, and put a matrix in reduced row echelon form

  • E3: I can compute the solution set for a system of linear equations or a vector equation.

• V: What is a vector space?

  • V1: I can explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn't a vector space.

  • V2: I can determine if a Euclidean vector can be written as a linear combination of a given set of Euclidean vectors by solving an appropriate vector equation.

  • V3: I can determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.

  • V4: I can determine if a subset of \(\IR^n\) is a subspace or not.

  • V5: I can determine if a set of Euclidean vectors is linearly dependent or independent by solving an appropriate vector equation.

  • V6: I can explain why a set of Euclidean vectors is or is not a basis of \(\IR^n\text{.}\)

  • V7: I can compute a basis for the subspace spanned by a given set of Euclidean vectors, and determine the dimension of the subspace.

  • V8: I can answer questions about vector spaces of polynomials or matrices.

  • V9: I can find a basis for the solution set of a homogeneous system of equations.

• A: How can we understand linear maps algebraically?

  • A1: I can determine if a map between vector spaces of polynomials is linear or not.

  • A2: I can translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.

  • A3: I can compute a basis for the kernel and a basis for the image of a linear map, and verify that the rank-nullity theorem holds for a given linear map.

  • A4: I can determine if a given linear map is injective and/or surjective.

• M: What algebraic structure do matrices have?

  • M1: I can multiply matrices.

  • M2: I can can express row operations through matrix multiplication.

  • M3: I can determine if a square matrix is invertible or not.

  • M4: I can compute the inverse matrix of an invertible matrix.

• G: How can we understand linear maps geometrically?

  • G1: I can describe how a row operation affects the determinant of a matrix.

  • G2: I can compute the determinant of a \(4\times 4\) matrix.

  • G3: I can find the eigenvalues of a \(2\times 2\) matrix.

  • G4: I can find a basis for the eigenspace of a \(4\times 4\) matrix associated with a given eigenvalue.