lecture 9 - linear mappings Flashcards
What is a mapping?
function f: E -> F where
E is a subset of R^n and F is a subset of R^m. U is a subset of E
f is a mapping if for all u in U, there exists some value v in F which directly maps to f(u) = w.
what is domain?
function f: E -> F where
E is a subset of R^n and F is a subset of R^m. U is a subset of E
the set U of elements that are mapped to F are the domain
represented as dom(f)
what is an image?
let f be mapping E -> F. The image of f are values w of set F which are mapped to from u where f(u) = w. a.k.a the set of vectors that have been transformed
The same as the range.
represented as Im(f)
example of image?
for R -> R f(x) = x^2 + 2x + 2
the domain is dom(f) = R
and image is Im(f) = [1, infinity]
imagine a quadratic graph but only the right positive part
what is injective?
“no two inputs give same output”
this means that no two vectors mapped to the same vector
> when rank of A equals the number of columns
> if A is square matrix, injectivity happens when det(A) != 0
(non-singular)
what is surjective?
“every possible output has an input”
> transformation covers entire output space
> if column space spans hte entire co-domain
rank(A) = number of rows
> if A is square matrix, surjectivity happens when det(A) != 0 (non-singular)
What is bijective (invertible)?
When function is both injective and surjective
> occurs only when square matrix has a full rank.
> det(A) != 0 (non-singular) so A is invertible
> Rank of A = number of rows = number of columns
What is a linear mapping?
A mapping that meets the requirements of:
additivity = if f(u + v) = f(u) + f(v) for all u, v of space E
homogeneity = if for all values of scalar λ that f(λv) = λf(v)
how are the linear mapping properties and vector space closures similar?
the L M properties are to ensure that when applying a linear mapping, it adheres to the closures inside of the vector space.
what proposition of linear mappings applies to the null vector?
if OE and OF are the null vectors of E and F vector spaces. linear mapping E > F follows that:
f(Oe) = oF
what is the matrix representation of a linear mapping?
a linear mapping that can be expressed as the product of a matrix and a vector
y = f(x) = Ax where A is a matrix
How do you get an image from a matrix?
Image of a matrix is the set that spans the column vectors of a matrix A.
so Im(f) = L(a1, a2, … an)
example of an image from a matrix
for f: R^3 > R^2 if A is:
1 -2 1
3 1 -1
the image is:
Im(f) = L((1,3),(-2,1),(1,-1))
What is an endomorphism?
> a mapping that maps to itself e.g. f: E -> E
a transformation that takes vectors and generates a new vector in the same dimensionality
square matrices
what is the kernel?
A set of elements v of vector space E who’s function is the null vector.
f(u - v) = Of (null vector)
u - v is an element of the kernel.
How is the kernel and injection related?
Mapping is injective IF and only IF ker(f) = {oE} (null vector)
e.g. if you take 2 different vectors and calculate their transformed vectors f(x,y,z) then you will not gain the same vector twice, so f is injective
this means that an injective mapping corresponds to a non-singular matrix, meaning that the kernel is only the null vector.
