8.1 General Linear Transformations Hwk: 1, 5, 9, 14, 17ab, 19, 21, 22, 28
Let u and v be vectors in a vector space V
and let k be any scalar. A linear
transformation is any function T:V-->W for
which the following two properties hold:
1. T(u + v) = T(u)
+ T(v) and
2. T(ku) = kT(u)
Note: u and v
are vectors in the domain V, and T(u), T(v)
etc. are vectors in the range W
+
8.2 Kernel and Range Hwk: 3, 4, 5a, 6a, 7bc, 8bc, 9, 10, 14ac, 16ac, 18, 19, 24, 25, 28, 29
New vocabulary for ideas developed previously
Kernel
Before: If A is an mxn matrix, the nullspace is set of all
solutions of the homogeneous system Ax=0, i.e. all the column vectors x (in Rn) that satisfy the
equation Ax = 0, i.e {x
| Ax = 0}
Also: The
nullspace of A is a subspace of Rn
The nullity of A is the
dimension of the nullspace (and the number of free variables in the rref)
Dimension theorem:
rank(A) + nullity(A) = n [the number of columns]
New: If T:V-->W (or TA),
is a linear transformation (or the lin trans corresponding to mult by the
matrix A), we say the kernel of T is the set of all vectors in V that
are mapped to the zero vector in W, that is {x
| T(x) = 0}
Also: The kernel of T is a subspace of V
The nullity of T is the dimension of the kernel, and nullity(TA) = nullity(A).
Rank(TA) = Rank(A)
Dimension theorem:
rank(T) + nullity(T) = n [dimension of the domain]
Range
Before: The column space of A is the set of
all vectors b in Rm for which there is at least one vector x in
Rn such that Ax = b, i.e. {b
| Ax = b for some x}
And: Column space of A is a subspace of Rm [the set of
vectors of length m]
New: The range of T:V-->W
is the set of all vectors in W which are the image of some vector in V, i.e. {b | T(x)
= b for some x in V}
And: Range(T) is a subspace of W
Some Tasks:
1. Is
a given vector v in the
kernel? We must have T(v) = 0.
So, calculate T(v).
To have v in ker(T), the result must be 0. Example:
Ex 1a vs 1b
2. Is
a given vector v in the
range? We must have T(x)= v for some unknown vector x.
So solve T(x)=v. If any solution exists (whether unique or
parameterized), v is in the range. Example:
Ex 2a vs 2b
3.
Find a basis for the kernel. This
is the same as finding a basis for the nullspace: solving T(x)=0 and
parameterizing the solution. Remember
that the trivial solution may be the only solution, in which case the kernel
consists of the 0 vector only.
4.
Find a basis for the range. If
given a matrix, find a basis for the column space. Otherwise, find the images of the standard
basis vectors. Any linearly independent
subset of these vectors will be a basis for the range.
8.3 Inverse Linear Transformations Hwk: 1, 2bc, 3d, 4ac, 5, 7, 8a, 9, 10ac, 11, 13, 17ac, 20
8.4 Matrices of General Linear Transformations Hwk: 1, 2, 3, 4, 7, 8, 11
8.5
Similarity
Return to: Merced College; Don Power Updated 05/11/07 by Don Power