Quiz 4 Flashcards
What is a function?
A function is a rule of correspondence between two sets, a domain and a range, that assigns each element in the domain exactly one element in the range. Important to note that not every element in the range necessarily needs to have a element in the domain which maps to it and not every element in the domain needs to map to a unique element in the range for these conditions to be fulfilled. See 3.1 for examples.
What is the image of a function?
The image of a function is the set of all elements in the range that are matched with elements in the domain by rule of correspondence
Can a relationship have an image if its not a function?
No
What is a transformation?
A transformation is a function with vector spaces for its domain and range
What is the degree of a polynomial?
The highest power of its variable
So t^3 + t^2 + t + 1 is a third degree polynomial
When is a transformation T linear?
For a transformation T:V–>W it is linear when the equality T(Au + Bv) = AT(u) + BT(v) for any two scalar A and B and any two vectors in V u and v
Generally we can think of T as being linear if it preserves linear combinations
A way to prove this is by proving T(u+v) = T(u) + T(v) and T(au) = aT(u) individually
What is a special case for linearity?
For a transformation L from R^n -> R^m, a matrix A which is mxn, and a vector u if L(u) = Au then L is linear
If V is a vector space and B is a basis for V with vectors {v1, v2, …., vn}, how do you write a vector v in V with respect to B?
If v = c1v1 + c2v2 + … + c3v3 then we can write v with respect to B as the n-tuple (the nx1 matrix) [c1 c2 c3 c4 … cn] _B
How can a LINEAR transformation be described completely and what does this mean?
A linear transformation can be described completely by its action on a basis for its domain
Since the transformation is linear, we know
T(Au + Bv) = AT(u) + BT(v)
holds.
Given this, and knowing how it acts on a basis for it’s domain, T(u) and T(v) can be how it acts upon its basis, and a and b can the coefficients required to make any vector in the domain out of the basis vectors.
What is not implied by a transformation?
Even if there is a transformation T:V->W that does not mean all elements in W can be accessed by the transformation of vectors in V. It simply means that the resulting transformed vectors are a subspace in W
What are some equivalencies for a matrix being invertible?
What is true about all linear transformations from one vector space to another?
They can be represented as a matrix where matrix is the basis vectors for the domain put through the transformation.
What is true of vectors?
They have a different representation, one for each basis of the vector space they are in
What is the transition matrix, how do you write it, and how do you find it?
The transition matrix is the matrix which transfer from some basis C to another basis D.
You write the transition matrix from C to D as (P_C)^D
You find it by writing each vector in the basis as a linear combination of the vectors which are a basis for D (aka you write the C vector into a new basis)
Then, with the transformed basis vectors, take each in a row, where the transformed v_1 vector is the 1st column of the transition matrix, and continue from there.
What is true about the transition matrix from C to D?
If C and D are finite dimension vector spaces, then the transition matrix is invertible, and its inverse is the transition matrix from D to C
