If T is any linear transformation which maps F^{n} to F^{m}, there is always an m×n matrix A ≡
 (2.23) 
for all x ∈ F^{n}. What is the form of A? Suppose T : F^{n} → F^{m} is a linear transformation and you want to find the matrix defined by this linear transformation as described in 2.23. Then if x ∈ F^{n} it follows

where e_{i} is the vector which has zeros in every slot but the i^{th} and a 1 in this slot. Then since Tis linear,


and so you see that the matrix desired is obtained from letting the i^{th} column equal T
Theorem 2.4.1 Let T be a linear transformation from F^{n} to F^{m}. Then the matrix A satisfying 2.23 is given by

where Te_{i} is the i^{th} column of A.
Proof: It remains to verify uniqueness. However, if A is a matrix which works, A =
Example 2.4.2 Determine the matrix for the transformation mapping ℝ^{2} to ℝ^{2} which consists of rotating every vector counter clockwise through an angle of θ.
Let e_{1} ≡
From Theorem 2.4.1, you only need to find Te_{1} and Te_{2}, the first being the first column of the desired matrix A and the second being the second column. From drawing a picture and doing a little geometry, you see that

Therefore, from Theorem 2.4.1,

Example 2.4.3 Find the matrix of the linear transformation which is obtained by first rotating all vectors through an angle of ϕ and then through an angle θ. Thus you want the linear transformation which rotates all angles through an angle of θ + ϕ.
Let T_{θ+ϕ} denote the linear transformation which rotates every vector through an angle of θ + ϕ. Then to get T_{θ+ϕ}, you could first do T_{ϕ} and then do T_{θ} where T_{ϕ} is the linear transformation which rotates through an angle of ϕ and T_{θ} is the linear transformation which rotates through an angle of θ. Denoting the corresponding matrices by A_{θ+ϕ}, A_{ϕ}, and A_{θ}, you must have for every x

Consequently, you must have

Don’t these look familiar? They are the usual trig. identities for the sum of two angles derived here using linear algebra concepts.
Example 2.4.4 Find the matrix of the linear transformation which rotates vectors in ℝ^{3}counterclockwise about the positive z axis.
Let T be the name of this linear transformation. In this case, Te_{3} = e_{3},Te_{1} =
 (2.24) 
In Physics it is important to consider the work done by a force field on an object. This involves the concept of projection onto a vector. Suppose you want to find the projection of a vector, v onto the given vector, u, denoted by proj _{u}

Because of properties of the dot product, the map v →proj _{u}
Example 2.4.5 Let the projection map be defined above and let u =
You can find this matrix in the same way as in earlier examples. proj _{u}

For the given vector in the example, this implies the columns of the desired matrix are

Hence the matrix is

Example 2.4.6 Find the matrix of the linear transformation which reflects all vectors in ℝ^{3} through the xz plane.
As illustrated above, you just need to find Te_{i} where T is the name of the transformation. But Te_{1} = e_{1},Te_{3} = e_{3}, and Te_{2} = −e_{2} so the matrix is

Example 2.4.7 Find the matrix of the linear transformation which first rotates counter clockwise about the positive z axis and then reflects through the xz plane.
This linear transformation is just the composition of two linear transformations having matrices

respectively. Thus the matrix desired is
