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L20100920 - From previous week Rank Example Solution From...

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Unformatted text preview: From previous week, Rank Example Solution From below shows that A basis of is How to solve this? Let , then __________________________________________________________________________________ Linear map (transformation) * field Q: Let Let Define How will the mapping look like? to be a basis of to be a basis of to be and for any linear combination Now, let Q: A: map. Why is it important to introduce linear map? Last time we introduce linear map before, because we can see something easier with linear The linear map, How do you see this one? Then we have Q: A: If you change basis of vector, L is fixed, but the associated A will change Then the associated matrix is Then Now let's change the basis Let Let So then, , still be the basis of , be the basis of It tells you that *Be careful here That means Further explanation on this, means the coordinate under a certain basis Let's say Let us look at P Let's be be will be will be * Look carefully it changes the axis Linear map won't change but the associated of the matrix will change. Why? Remember map can't change, that’s why the vector changes. If coordinate is it will become The point is fixed but the axis switch around so the coordinate change. So go back where * Just the point is fixed. CONFUSING!!!! Q: A: What is the reason for changing basis? to get the simplest A We can see that if you change the basis your A will change Q: A: So, what is the simplest A here? First of all, you can't get identity matrix in and *Notice that To calculate We want to get a simplest matrix Because is are all fixed! Q: Linear map implies something about matrix, same goes for matrix. So choose a basis in a linear map, what does it mean in the matrix? A: Goal: Changing basis of Row and Column operations and , such that the associated is as simple as possible. Find the appropriate basis in Make the appropriate Row or Column operation to the original , such that Example: is simple Therefore , Row operation Column operation Column operation ...
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