Section 1.8 Intro to Linear Transformations
Range of T
1. Set of all images 2. The images of all x in the domain under T
Proving Linear Transformation
1. T(0) = 0 2. T(CU) = CT(U) 3. T(U+V) = T(U) + T(V)
If A is a 3x5 matrix and T is a transformation then the domain of T is R^3
False R^n so R^5
If T is a linear trans and if c is in r^m, then a uniqueness question is "is c in the range of T?"
False that's a uniqueness question
If A is an mxn matric then the range of T is R^m
False that's the codomain. the range is the set of all linear combinations of the columns of A
Every linear transformation is a matrix transformation
False, reverse it.
Is b in the range of the linear transformation? What other way is there to say this?
Is b in the span of the columns of A or is [A | b] consistent?
One to one
Pivot in every column Columns of A are linearly independent Only trivial solution No free variables
If c1*T(v1) + c2*T(v2) + c3*T(v3) = 0 and T is linear what does this do?
T(c1v1+c2v2+c3v3)
Linear transformation if
T(cu+dv) = cT(u) + dT(v)
T(c1v1+c2v2+c3v3) what happens if T is one to one?
T(x) = 0 implies x = 0 so gets rid of T c1v1+c2v3+c3v3
What is the image of x under T?
T(x) is called the image of x under T for a vector x in domain
A linear transformation is a special type of function
True
A linear transformation preserves the operations of vector addition and scalar multiplication
True
Every matrix transformation is a linear transformation
True
The range of T is the set of all linear combinations of the columns of A
True
A linear trans always maps the origin of r^n to the origin of r^m
True t(0) = 0
T is determined by what?
What T does to the columns of an nxn identitymatrix
Range
What actually comes out of a function. The set of all images T(x). Or the set of all linear combinations of the columns of A.
Domain
What can go into a transformation. The set in R^n in T. Or the number of columns
Co domain
What can possibly come out of a transformation. The set in R^m. Or the number of rows in each column
The question for transformations is what?
What does T do to the vector given
If T maps r^n to r^m is linear transformation then does there exist a unique standard matrix?
Yes
Is every matrix transformation a linear transformation?
Yes
Can every linear transformation be defined as a matrix multiplication?
Yes for every T : Rn −→ Rm
If T is linear then is T(0) = 0?
Yes since 0 * T(u)= 0
onto
if each b in codomain is the image of at least one x in domain. Or Pivot in every row of A so columns of A span r^m
Is b in range? What to look for
is [A | b] consistent if so then it's in range
The image of a linear combination is the
linear combo of the image of each vector under T