Pure Mathematics C3: Linear Transformations

34 flashcards · Mathematics · by lakesare

1. When is a function called a transformation?

   Transformation is function that maps R2 → R2. 4 examples: dilation, scaling, rotation and reflection.

2. How do we define a linear transformation?

   t: V → W that satisfies: LT1: t(v1 + v2) = t(v1) + t(v2) for all v1, v2 ∈ V.LT2: t(αv) = αt(v) for all v ∈ V , α ∈ R.

3. What makes linear transformations important? Suppose that t: V → W is a linear transformation.

   Because once we know the images of each vector in a basis for V, we can find the image of all vectors in V.

4. What's the strategy to determine whether t is a linear transformation?

5. [Linear transformations] What is a zero transformation?

6. [Linear transformations] What is an identity transformation?

   We omit the subscript V when the vector space is clear from the context.

7. [Linear transformations] Why is if t is a linear transformation, then t(0) = 0?

   Because of the LT2 property:

8. [Linear transformations] Visualise LT1 property.

   Parallelograms are mapped to parallelograms

9. [Linear transformations] Visualise LT2 property.

   Scalar multiples are preserved

10. Linear combination VS linear equation VS linear transformation?

    linear combination linear equation linear transformation

11. What does it mean to say that linear transformation "preserves" vector addition? Vector spaces have two operations: vector addition and scalar multiplication.

    Preserving addition means that we get the same result whether we add first and then transform, or transform first and then add.

12. Why is this kind of transformation called "linear transformation"? linear combination linear equation linear transformation

    Because it preserves linear combinations!

13. Why is it important that t preserves linear combinations? Linear combinations of any number of vectors are preserved under a linear transformation.

    Because given t, and given images of vectors in a basis for V, we can find the images of all vectors in V! This method of finding an algebraic definition for rθ is simpler than the geometric approach, and is more generally applicable.

14. [E-coordinate representation] Describe this in language.

    v1, ...vn are the coordinates of v with respect to the basis { e1, ...en }.

15. Imagine visually - what's the desired result of finding matrix representation of t with respect to bases E (domain) and F (codomain)?

16. How do matrices fit into all this? linear combination linear equation linear transformation

    Function f is a linear transformation ≡ function f has a matrix representation. Similar expressions exist for linear transformations from Rn to Rm.

17. What does it mean "with respect to the bases E and F"?

    "With respect to the basis E for domain, and basis F for codomain".

18. How would we find the inverse of the linear transformation?

    We would make use of its matrix representation A, and just find A–1.

19. Why if t is invertible, then dim(V) = dim(W)?

    Because only square matrices can be invertible.

20. [Isomorphism between vector spaces] When are 2 groups isomorphic VS 2 vector spaces isomorphic?

    2 groups - when there exists φ: G → H such that: φ is one-to-one and onto φ(x ◦ y) = φ(x) ∗ φ(y) 2 vector spaces - when there exists t: V → W such that: t is invertible The function is invertible iff it is one-to-one and onto.t is a linear transformation You met isomorphisms between groups in Unit B2 Subgroups and isomorphisms. Isomorphisms between vector spaces are analogous: they identify when vector spaces are ‘structurally identical’ to each other.

21. [Isomorphism between vector spaces] Write down some isomorphism from P3 to R3.

    (There are many other possibilities.)

22. [Isomorphism between vector spaces] What's the easiest way to determine whether 2 vector spaces are isomorphic?

    V and W are isomorphic ≡ dim V = dim W.

23. What does "Im t" stand for?

    Image set of a linear transformation t.

24. What do "Im t" VS "Im z" likely mean?

    Im t - Image set of a linear transformation t. Im z - Imaginary part of a complex number z.

25. Name 2 steps to find the image set of any linear transformation t: V → W. Once we know a basis for the image set of a linear transformation, we know everything that we need to know about the image se For the linear transformations studied so far, it has been easy to write down their image sets. In general, this is not the case; so we need some way of determining the image set of a linear transformation.

    Find a basis for the domain { e1, ...en }, determine the vectors { t(e1), ...t(en) } Remove all linearly dependent vectors!

26. What's the kernel of the linear transformation? Lev Semyonovich Pontryagin

    All vectors in the domain that transform into 0.

27. Visually imagine the kernel of t.

28. Crack a handful of whole new season's walnuts, remove the kernels from the shells and halve and quarter them.

    A softer, usually edible part of a nut, seed, or fruit stone contained within its shell.

29. [Solution Set Theorem] Visually imagine the solution set of this equation. We want to find x.

30. [Solution Set Theorem] What's the solution set?

    { x : x = a + k for some k ∈ Ker(t) }

31. [Solution Set Theorem] What's a in the Solution Set Theorem?

    a is some vector in V that does map to b.

32. Imagine both Ker t and Im t in the same image.

    The subspaces Ker t and Im t

33. [Dimension Theorem] State the Dimension Theorem. The subspaces Ker t and Im t

    dim(Im t) + dim(Ker t) = dim V

34. What important result does this theorem: Dim V VS Dim W ⟶ t is one-to-one/onto lead to?

    Theorem: n of linear equations VS n of unknowns ⟶ n of solutions.