Linear Algebra
Students are often so caught up in the numbers in matrices, and the calculations, that they lose sight of what is going on geometrically. The aim of the animations on this page is to provide phenomena, which, when accounted for, rehearse the main ideas in linear algebra.
After watching an animation, tell yourself a story about what is going on, for example, as a commentary that could be inserted in the animation..
Keep it simple at first; only resort to technical terms when you are confident you comprehend what is going on.
The Cinderella applets (stand alone) used to generate the animations can be downloaded and used for personal exploration.
Applet for animations 1 through 4
Applet for Change of Basis
After watching an animation, tell yourself a story about what is going on, for example, as a commentary that could be inserted in the animation..
Keep it simple at first; only resort to technical terms when you are confident you comprehend what is going on.
The Cinderella applets (stand alone) used to generate the animations can be downloaded and used for personal exploration.
Applet for animations 1 through 4
Applet for Change of Basis
Linear Transformations
Animation 1: Coordinates
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What is going on here?
Construct a narrative (perhaps a possible sound track) telling viewers what they are seeing. What questions seem to be posed at the end? |
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Animation 2: Tracking the image of a circle
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What is going on here?
Watch out for extra vectors, a marked angle, and the cosine of that angle (tracked in green). Construct a narrative (perhaps a possible sound track) telling viewers what they are seeing. What further questions does this animation raise? |
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Animation 3: The effect on the image of a circle of varying the transformation
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Holding f2 fixed, what is the boundary of the region in which f1 lies so that there are real eigenvectors?
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Animation 4: column and row spaces
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What is going on here?
What relationships are there between the row and column spaces? Both spaces may be helpful in making full sense of animation 2 |
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Change of Basis
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What is going on here?
How does it relate to a change of basis? |
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Further Questions for Exploration
1. The set of matrices mapping V to V .which have a fixed vector v as an eigenvector, is closed under addition and multiplication, and scalar multiplication. Characterise the subset of these which are zero-divisors in the ring.
2. Given a non-singular linear transformation T of V to V, what is the set of matrices which can be used to represent T?
2. Given a non-singular linear transformation T of V to V, what is the set of matrices which can be used to represent T?