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Theorem 10

If T is a linear transformation, then there exists a unique matrix A such that T(x) = Ax for all x in R ^n.

standard matrix for the linear transformation T

The matrix A in [T(e1) ... T (en)] x = Ax

geometric linear transformations

determined by what they do to the columns of I2; 1. reflections (over axes/ lines); 2. contractions/ expansions (size changes); 3. shears (stretching diagonally); 4. projections

mapping T onto R^ m

each b in R^ m is the image of at least one x in R^n; (consistency)

mapping T one to one

each b in R^ m is the image of at most one x in R^ n; (uniqueness)

Theorem 11

If T is a linear transformation, then T is one to one iff the equation T (x) = 0 has only the trivial solution (linearly independent set)