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This session is a reserve slot used for interactive problem solving. It covers two worked problems: (P1) checking invertibility, finding eigenvalues/eigenvectors, and diagonalising a 2×2 matrix via a basis transformation; (P2) analysing a projection transformation — finding its matrix representation, nullspace, eigenvalues, and eigenvectors, with geometric interpretation. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Determine invertibility by computing $\det(A) \neq 0$
• [ ] Find eigenvalues by solving $\det(A - \lambda I) = 0$
• [ ] Find eigenvectors by solving $(A - \lambda I)\vec{v} = 0$ for each eigenvalue
• [ ] Construct the basis transformation matrix $M$ from eigenvectors as columns
• [ ] Compute $M^{-1}$ using Gaussian elimination on $(M|I)$
• [ ] Diagonalise: $A' = M^{-1}AM$ and verify the result is $\text{diag}(\lambda_1, \lambda_2)$
• [ ] Verify $M^{-1}M = I$ as a check
• [ ] Derive the matrix representation of a projection from $T\vec{v} = \langle \vec{v}, \hat{u} \rangle \hat{u}$
• [ ] Find the nullspace of a transformation by solving $T\vec{v} = 0$
• [ ] Find eigenvalues/eigenvectors of a projection and interpret geometrically
• [ ] Understand that $\lambda = 0$ means the eigenvector is in the nullspace
• [ ] Understand that $\lambda = 1$ for a projection means the eigenvector is along the projection direction

1. Problem 1: Invertibility and Diagonalisation

The Matrix

$$ A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$

Part (a): Is $A$ Invertible?