Section 12 Week 4 Learning Goals

Here are the knowledge and skills you should master by the end of the fourth week.

12.1 Vector Spaces and the Determinant

I should be able to do the following tasks:

  • Prove/disprove that a subset of a vector space is a subspace.
  • Prove/disprove that a set of vectors is linearly dependent.
  • Prove/disprove that a set of vectors span a vector space (or a subspace).
  • Find the kernel and image of \(T(\mathsf{x}) = Ax\).
  • Determine whether a set of vectors is a basis.
  • Find a basis for \(\mathrm{Nul}(A)\) and a basis for \(\mathrm{Col}(A)\).
  • Find the change-of-coordinate matrix \(P_{\mathcal{B}}\) from basis \({\mathcal{B}}\) to the standard basis \(\mathcal{S}\).
  • Use matrix inverses (and RStudio) to find the change-of-coordinate matrix \(P_{\mathcal{B}}^{-1}\) from basis \({\mathcal{S}}\) to the standard basis \(\mathcal{B}\).
  • Find the coordinate vector with respect to a given basis.
  • Find the dimension of a vector space (or subspace) by finding or verifying a basis.
  • Find the determinant of a \(2 \times 2\) matrix by hand.
  • Find the determinant of a \(3 \times 3\) matrix by using row operations/cofactor expansion/permutation method.
  • Use RStudio to calculate the determinant of a square matrix.
  • Use \(\det(A)\) to decide whether the square matrix \(A\) is invertible.

12.2 Vocabulary

I should know and be able to use and explain the following terms or properties.

  • every one of these Important Definitions
  • subspace
  • null space and column space of a matrix
  • kernel and image of a linear transformation
  • basis
  • coordinate vector with respect to a basis
  • change-of-coordinates matrix
  • the coordinate vector with respect to a basis
  • the dimension of a vector space (or a subspace)
  • determinant

12.3 Conceptual Thinking

I should understand and be able to explain the following concepts:

  • A vector space consists of a collection of vectors and all of their linear combinations.
  • A subspace is a subset of a vector space that is also a vector space by itself (closed under linear combinations).
  • The solutions to \(A \mathsf{x} = \mathbb{0}\) form a subspace.
  • The span of the columns of \(A\) form a subspace.
  • How the kernel and image of \(T(\mathsf{x}) = Ax\) correspond to the nullspace and columnspace of \(A\).
  • Every basis of a given vector space (or subspace) contains the same number of vectors.
  • Why every vector in a vector space has a unique representation as a linear combination of a given basis \({\mathcal{B}}\).
  • How dimension relates to span and linear independence.
  • Interpret \(\det(A)\) as a measure the expansion/contraction of “volumes” in \(\mathbb{R}^n\) under the linear transformation \(T(\mathsf{x})=A\mathsf{x}\).