Change of Basis (using coordinate isomorphism)

Linear Algebra Pt.22

0:00 Matrix of Transformation with Standard Basis of R2
0:44 Matrix of Transformation with Standard Basis of P2
0:56 Intro to Coordinate Isomorphism
2:07 Matrix of Transformation with Standard Basis of P2 Continued...
3:07 Ex - Find coordinates of matrix in terms of basis B
4:29 Commutative Diagram
6:22 Ex - Find the Transformation (given matrix of T in terms of non-standard bases)
8:59 Ex - The Plan: Find the matrix of T in terms of B and D (non-standard bases)
9:32 Ex - Method #1: Find the matrix of T in terms of B and D (non-standard bases)
11:05 Ex - Method #2: Find the matrix of T in terms of B and D (non-standard bases)

Linear Algebra Playlist:    • Linear Algebra  

1. Vector Space Walkthrough & examples    • Vector Space Axioms Walkthrough & examples  
2. Is it a Vector Space? (featuring crazy addition and multiplication rule)    • Is it a Vector Space? (featuring crazy add...  
3. Extra Vector Space Properties & Proofs    • Extra Vector Space Properties & Proofs  
4. Subspace Test Shortcut (and examples)    • Subspace Test Shortcut (and examples)  
5. Span! Why is it the smallest subspace? When are 2 spans equal?    • Span! Why is it the smallest subspace? Whe...  
6. Linearly Independent or Dependent?    • Linearly Independent or Dependent?  
7. Basis, Standard Basis, Dimension? (& finding multiple bases)    • Basis, Standard bases, Dimension? (& findi...  
8. Carving & Enlarging Set to find a Basis (with the independent + dependent lemma)    • Carving & Enlarging Set to find a Basis (w...  
9. Row vs Column space of a matrix    • Row vs Column space of a matrix  
10. What’s a LINEAR Transformation? Linear Operator? Matrix of a Transformation?    • What's a LINEAR transformation? Linear ope...  
11. Kernel, Image & Dimension of Transformations Explained!    • Kernel, Image & Dimension of Transformatio...  
12. Injectivity, Surjectivity & Bijectivity made easy    • Injectivity, Surjectivity & Bijectivity ma...  
13. Isomorphisms? Coordinate Isomorphisms? Proving linearity & bijection!    • Injectivity, Surjectivity & Bijectivity ma...  
14. Composition of linear transformations, Finding T INVERSE    • Composition of linear transformations, Fin...  
15.Orthogonality? Normalization? Expansion Theorem!    • Orthogonality? Normalization? Expansion Th...  
16. Converting to an Orthogonal Basis! How “nice”    • Converting to an Orthogonal Basis! How "nice"  
17. Orthogonal Projections (featuring U + U perp)    • Orthogonal Projections (featuring U + U perp)  
18. Similar Matrices, Properties & Proofs    • Similar Matrices, Properties & Proofs  
19. Diagonalization (And what actually are Eigenvalues, Eigenvectors and Eigenspaces?)    • Diagonalization (And what actually are Eig...  
20. Showing Similarity when A~D and B~D    • Showing Similarity when A~D and B~D  
21. Orthogonal Diagonalization    • Orthogonal Diagonalization  
22. Change of Basis (using coordinate isomorphism)    • Change of Basis (using coordinate isomorph...