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
What is true of a linear transformation?
It has many matrix representation, generally one for each pair of bases in the domain and range
What is true of two nxn matrices A and B?
They represent the same linear transformation if and only if there exists an invertible matrix P such that
A = P^-1BP
When are two matrices similar?
When they represent the same linear transformation, in which case there exists A = P^-1BP
How do you write the representative matrix for a transformation?
Plug in basis vectors for the domain into the transformation, taking the result, representing it relative to the given range, and having the 1st basis vector transformation be the first column, the 2nd basis vector transformation be the second column, and so on
When a transformation T is linear what is true of the zero vector?
T(0) = 0
What is the kernel of a linear transformation T?
The set of all vectors v such that T(v) = 0
What is the image of a linear transformation T?
The set of all vectors w such that for some v in the domain
T(v) = W
What are the nullity and rank of a linear transformation?
The dimension of its kernel and its image respectively
What does r(T) and v(T) represent?
The rank and the nullity of the linear transformation T
What is true of r(T) and v(T)?
r(T) + v(T) = n where n is the dimension of the domain
What is true of a one to one transformation?
Each vector in V has a unique vector in W which it maps to
What is true if and only if the kernel only contain the 0 vector?
The linear transformation is one to one
When is a linear transformation onto?
When its image is its range
What is true if for a transformation T:V—>W if V and W have the same dimension?
T is onto if and only if T is one to one
