Everything so far Flashcards
What is the definition of the inverse of a function?
A function g : Y-> X is the inverse of f : X->Y if and only if both of these statements are true:
1. g(f(x))=x for all x in X
2. f(g(y))=y for all y
Image of a linear map
Im(A) = the set of A(x) for all x in X
(Basically, the range)
Surjective
All y in Y can be reached by some A(x)
Injective
If x1 != x2, f(x1) != f(x2)
And vice versa
If a set contains ___, it is automatically dependent.
Why?
0; because then you could have 0v + 10 = 0 (a linear combination other than the 0 vector of the things)
The image of a linear map is ___. Why?
the column space.
this is because:
1. multiplying a matrix by a vector (i.e. applying the map to an input vector) can be seen as taking x of first col of matrix, y of second col of matrix, z of third col of matrix, etc.
2. we are taking the span of this, considering all possible x y z you can take (hence all possible outputs.)
A linear map is surjective if and only if ___.
The image is the same is the codomain.
The linear map/transformation of 3x2 matrix maps from…
R2 to R3
What does it mean to apply a linear transformation from one space to another?
Example. Consider taking in R1 and mapping to R3. You could just take the input multplied by the vector (1 2 3) for example.
What are the two ways to see matrix multiplication?
- Take each row, multply first entry in row by first entry in input, etc.
- for input vector ( x, y, z ): take x of first col, y of second col, z of third col
rank of a matrix
the dimension of the column space (how many vectors you need to span the entire column space)
Kernel of a linear transformation
The set of all input x in X such that f(x) = 0.
preimage
for some given y in Y, the set of all x such that f(x) = y.
For any SURJECTIVE map, it must be the case that #Y _ #X
Y <= #X
For any INJECTIVE map, it must be the case that #Y _ #X
Y >= #X
