1.9 the matrix of a linear trnasformation

The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form [a 0][0 d]

True

Why is the question "is the linear transformation T onto?" an existence question?

If T is onto, then there is at least one vector x in the domain of T, for every vector y in codomain of T so that T(x) = y. In other words, solutions exist for every right side Tx = y.

If a linear transformation T: Rn -> Rm maps Rn onto Rm, can you give a relation between m and n? If T is one to one, what can you say about m and n?

If T: Rn -> Rm maps Rn onto Rm, then its standard matrix A has a pivot in each row. A must have at least as many columns as rows. That is, n <= m. When T is one to one, A must have a pivot in each column so m >= n

one-to-one

A mapping T: Rn -> Rm is said to be one to one if each Rm is the image of at most one x in Rn - T is one to one if for each b in Rm, the equation T(x) = b has either a unique solution or none at all - T is one to one if and only if the columns of A are linearly independent

onto

A mapping T: Rn -> Rm is said to be onto Rm if each b in Rm is the image of at least one x in Rn - T maps Rn onto Rm if and only if the columns of A span Rm

The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n x n identity matrix

True, the standard matrix is the mxn matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rn

If T: R2-->R2 rotates vectors about the origin through an angle z, then T is a linear transformation

True, the standard matrix, A, of the linear transformation is [cos z -sinz][sinz cosz]

A linear transformation T: Rn -> Rm is completely determined by its effect of the columns of the nxn identity matrix

True, the vector x can be written as a linear combination of the columns of the identity matrix. T is a linear transformation so T(x) can be written as a linear combination of the vectors T(e_1) through T(e_n)

If A is a 3x2 matrix, then the transformation x->Ax cannot map R2 onto R3

True, you cant map a space of lower dimension onto a space of higher dimension

A mapping T: Rn --> Rm is onto Rm if every vector x in Rn maps onto some vector in Rm

False, a linear transformation is onto if the codomain is equal to the range. A mapping T: Rn -> Rm is onto Rm if every vector in Rm is mapped onto by some vector x in Rn

A mapping T: Rn-->Rm is one to one if each vector in Rn maps onto a unique vector in Rm

False, a mapping is one to one if each vector in Rm is mapped to form a unique vector in Rn

When 2 linear transformations are performed one after another the combined effect may not always be a linear transformation

False, a transformation is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v, and scalars c. The first transformation results in some vector u, so the properties of a linear transformation must still apply when 2 transformations are applied

Not every linear transformation from Rn to Rm is a matrix transformation.

False, for a linear transformation from Rn to Rm we see where the basis vector in Rn gets mapped to. These form the standard matrix

If A is a 3x2 matrix, then the transformation x->Ax cannot be one-to-one.

False, since the transformation maps from R2 to R3 and 2 < 3, it can be one to one but not onto. A transformation is one to one if each vector in the codomain is mapped to by at most one vector in the domain. If Ax = b does not have a free variable, then the transformation represented by A is one to one

Let T:Rn-->Rm be a linear transformation and let A be the standard matrix for T. T maps Rn onto Rm if and only if A has _________ pivot columns

m pivot columns; columns of A must span Rm which is only possible if A has a pivot position in each row

Let T:Rn-->Rm be a linear transformation and let A be the standard matrix for T. T is one to one if and only if A has ____________ pivot columns

n pivot columns; all columns of A are only linearly independent if A has n pivot columns