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This lecture connects several big ideas: how to transform coordinates between different bases using the basis transformation matrix, why orthonormal bases make finding coordinates trivially easy (just take scalar products!), why matrix–vector multiplication is row × column, and how to transform matrices between coordinate systems via similarity transformations. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Construct the basis transformation matrix $M$ by gluing basis vectors as columns
• [ ] Transform coordinates: basis $B$ → standard via $M$, standard → basis $B$ via $M^{-1}$
• [ ] Work through the full example: find $(5/3, 1/3)^T$ as coordinates of $(3,2)^T$ in basis $b_1=(2,1)^T$, $b_2=(-1,1)^T$
• [ ] Know how to verify a coordinate transformation result
• [ ] For an orthonormal basis, find coordinates via scalar products: $v_j = \langle v, u_j \rangle$
• [ ] Prove this using the Kronecker delta property $\langle u_i, u_j \rangle = \delta_{ij}$
• [ ] Work through the orthonormal example: $v = (3,1)^T$ in basis $u_1 = \frac{1}{\sqrt{2}}(1,1)^T$, $u_2 = \frac{1}{\sqrt{2}}(-1,1)^T$
• [ ] Derive why matrix–vector multiplication is row × column from linear maps + scalar products
• [ ] State the cookbook recipe: apply $T$ to all basis vectors, stack results as columns → matrix
• [ ] Define the similarity transformation: $A_{\text{red}} = M^{-1} A_{\text{black}} M$
• [ ] Work through the permutation matrix example in a non-standard basis

1. Basis Transformation Matrix

The Problem

We can have many different bases for the same vector space. Given a vector in one basis, how do we find its coordinates in another?

From a Non-Standard Basis to the Standard Basis (Easy Direction)

Suppose we have basis $B = \{b_1, b_2\}$ with $b_1 = (2,1)^T$ and $b_2 = (-1,1)^T$.