The standard

matrix of a

linear transformation is a matrix that

induces the transformation. Properties of this matrix
will imply properties of the linear transformation itself.

To find the standard matrix of a linear transformation, simply construct a
matrix whose columns are the output of the transformation when applied to the
standard
vectors *e*_{1}...*e*_{n}
(where *n* is the dimension of the transformation's domain).

**Theorem:**

Let T:**R**^{n}→**R**^{m} be a linear
transformation. Then there exists a unique matrix A, called *the **standard
matrix* of T, such that T = T_{A}.

**Proof:**

**Existence:**

Let A be m x n matrix whose columns are a_{j} =
T(e_{j}). Then:
T_{A}(e_{j}) = Ae_{j} = a_{j}

T_{A} = T

**Uniqueness:**

If T_{A}(e_{j}) = T_{B}(e_{j}), then A = B.

**Properties:**

Consider the linear transformation
T:**R**^{n}→**R**^{m} and its standard matrix
A∈M(m,n):

The statements within each of the following groups are equivalent; the groups
are not equivalent to each other, nor are statements that exist in separate
groups:

- T is onto.
- The equation A
**x** = b has at least 1 solution for any given b.
- The columns of A are a spanning set for
**R**^{m}.
- rank(A) = m

- T is invertible
- The equation A
**x** = b has exactly one solution for any given b.
- The columns of A are linearly independent and span
**R**^{m}.
- rank(A) = m = n

**Example:**
Well,

MathML doesn't appear to be very widely supported. If it was, there'd be
matrices all over the place down here, but, well... Sorry.

*Feel free to drop me a note if this changes and I don't notice.*