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Appendix G Figures and video examples

0.5 Proof techniques

Figure 0.5.9 Mathematical induction as ladder of propositions

1.1 Vector space structure of \(\R^n\)

Figure 1.1.17 Point visualization of triple

Figure 1.1.18 Vector visualization of triple

Figure 1.1.19 Tip-to-tail visualization of vector addition

Figure 1.1.20 Tip-to-tip visualization of vector difference

Figure 1.1.21 Tip-to-tail visualization of vector addition

2.1 Systems of linear equations

Figure 2.1.8 Using Sage to visualize \(\mathcal{P}\colon ax+by+cz=d\) via normal vector \(\boldn=(a,b,c)\)

2.2 Gaussian elimination

2.3 Solving linear systems

Figure 2.3.8 Video: solving linear systems

Figure 2.3.10 Video: solving linear systems 2

Figure 2.3.12 Decision tree for number of solutions to a system

3.1 Matrix arithmetic

Figure 3.1.22 Visualizing matrix multiplication

3.2 Matrix algebra

3.4 The invertibility theorem

3.5 The determinant

4.1 Subspaces

4.2 Span and linear independence

4.3 Bases

4.4 Dimension

4.5 Fundamental spaces

Figure 4.5.4.(a)

Figure 4.5.4.(b)

5.1 Linear transformations

Figure 5.1.19 Video: deciding if \(T\) is linear

Figure 5.1.20 Video: deciding if \(T\) is linear

Figure 5.1.40 Visualizing reflection and rotation

5.2 Null space, image, and isomophisms

Figure 5.2.2 Null space and image

5.3 Coordinate vectors

5.4 Matrix representations of linear transformations

Figure 5.4.8 Commutative diagram for \([T]_B^{B'}\)

5.5 Change of basis

Figure 5.5.24 The holy commutative tent of linear algebra

6.1 Eigenvectors and eigenvalues

Figure 6.1.2 Reflection through \(\ell=\Span\{(1,2)\}\)

Figure 6.1.7 Visualizing eigenvectors

6.2 Diagonalization

Figure 6.2.16 Video: deciding if diagonalizable

7.3 Orthogonal projection

Figure 7.3.14 Orthogonal projection onto a line

Figure 7.3.18 Orthogonal projection onto plane and normal line

Figure 7.3.30 Least-squares visualization

7.4 The spectral theorem

Figure 7.4.14 Eigenspaces of a symmetric matrix are orthogonal