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