| Term | Definition |
|---|
| 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) |
| Theorem 12 | Let T be the linear transformation and let A be the standard matrix for T then: a. T maps R^n onto R^m iff the columns of A span R^ m and b. T is one to one iff the columns of A are linearly independent |
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