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Mathematics LibreTexts

5.2: The Matrix of a Linear Transformation I

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Outcomes

  1. Find the matrix of a linear transformation with respect to the standard basis.
  2. Determine the action of a linear transformation on a vector in Rn.

In the above examples, the action of the linear transformations was to multiply by a matrix. It turns out that this is always the case for linear transformations. If T is any linear transformation which maps Rn to Rm, there is always an m×n matrix A with the property that T(x)=Ax for all xRn.

Theorem 5.2.1: Matrix of a Linear Transformation

Let T:RnRm be a linear transformation. Then we can find a matrix A such that T(x)=Ax. In this case, we say that T is determined or induced by the matrix A.

Here is why. Suppose T:RnRm is a linear transformation and you want to find the matrix defined by this linear transformation as described in (???). Note that x=[x1x2xn]=x1[100]+x2[010]++xn[001]=ni=1xiei where ei is the ith column of In, that is the n×1 vector which has zeros in every slot but the ith and a 1 in this slot.

Then since T is linear, T(x)=ni=1xiT(ei)=[||T(e1)T(en)||][x1xn]=A[x1xn] The desired matrix is obtained from constructing the ith column as T(ei). Recall that the set {e1,e2,,en} is called the standard basis of Rn. Therefore the matrix of T is found by applying T to the standard basis. We state this formally as the following theorem.

Theorem 5.2.2: Matrix of a Linear Transformation

Let T:RnRm be a linear transformation. Then the matrix A satisfying T(x)=Ax is given by A=[||T(e1)T(en)||] where ei is the ith column of In, and then T(ei) is the ith column of A.

The following Corollary is an essential result.

Corollary 5.2.1: Matrix and Linear Transformation

A transformation T:RnRm is a linear transformation if and only if it is a matrix transformation.

Consider the following example.

Example 5.2.1: The Matrix of a Linear Transformation

Suppose T is a linear transformation, T:R3R2 where T[100]=[12], T[010]=[93], T[001]=[11] Find the matrix A of T such that T(x)=Ax for all x.

Solution

By Theorem 5.2.2 we construct A as follows: A=[||T(e1)T(en)||]

In this case, A will be a 2×3 matrix, so we need to find T(e1),T(e2), and T(e3). Luckily, we have been given these values so we can fill in A as needed, using these vectors as the columns of A. Hence, A=[191231]

In this example, we were given the resulting vectors of T(e1),T(e2), and T(e3). Constructing the matrix A was simple, as we could simply use these vectors as the columns of A. The next example shows how to find A when we are not given the T(ei) so clearly.

Example 5.2.2: The Matrix of Linear Transformation: Inconveniently Defined

Suppose T is a linear transformation, T:R2R2 and T[11]=[12], T[01]=[32] Find the matrix A of T such that T(x)=Ax for all x.

Solution

By Theorem 5.2.2 to find this matrix, we need to determine the action of T on e1 and e2. In Example 9.9.2, we were given these resulting vectors. However, in this example, we have been given T of two different vectors. How can we find out the action of T on e1 and e2? In particular for e1, suppose there exist x and y such that [10]=x[11]+y[01]

Then, since T is linear, T[10]=xT[11]+yT[01]

Substituting in values, this sum becomes T[10]=x[12]+y[32]

Therefore, if we know the values of x and y which satisfy (???), we can substitute these into equation (???). By doing so, we find T(e1) which is the first column of the matrix A.

We proceed to find x and y. We do so by solving (???), which can be done by solving the system x=1xy=0

We see that x=1 and y=1 is the solution to this system. Substituting these values into equation (???), we have T[10]=1[12]+1[32]=[12]+[32]=[44]

Therefore [44] is the first column of A.

Computing the second column is done in the same way, and is left as an exercise.

The resulting matrix A is given by A=[4342]

This example illustrates a very long procedure for finding the matrix of A. While this method is reliable and will always result in the correct matrix A, the following procedure provides an alternative method.

Procedure 5.2.1: Finding the Matrix of Inconveniently Defined Linear Transformation

Suppose T:RnRm is a linear transformation. Suppose there exist vectors {a1,,an} in Rn such that [a1an]1 exists, and T(ai)=bi Then the matrix of T must be of the form [b1bn][a1an]1

We will illustrate this procedure in the following example. You may also find it useful to work through Example 5.2.2 using this procedure.

Example 5.2.3: Matrix of a Linear Transformation Given Inconveniently

Suppose T:R3R3 is a linear transformation and T[131]=[011],T[011]=[213],T[110]=[001] Find the matrix of this linear transformation.

Solution

By Procedure 5.2.1, A=[101311110]1 and B=[020110131]

Then, Procedure 5.2.1 claims that the matrix of T is C=BA1=[224001436]

Indeed you can first verify that T(x)=Cx for the 3 vectors above:

[224001436][131]=[011], [224001436][011]=[213] [224001436][110]=[001]

But more generally T(x)=Cx for any x. To see this, let y=A1x and then using linearity of T: T(x)=T(Ay)=T(iyiai)=yiT(ai)yibi=By=BA1x=Cx

Recall the dot product discussed earlier. Consider the map v proju(v) which takes a vector a transforms it to its projection onto a given vector u. It turns out that this map is linear, a result which follows from the properties of the dot product. This is shown as follows. proju(kv+pw)=((kv+pw)uuu)u=k(vuuu)u+p(0.05inwuuu)u=kproju(v)+pproju(w)

Consider the following example.

Example 5.2.4: Matrix of a Projection Map

Let u=[123] and let T be the projection map T:R3R3 defined by T(v)=proju(v) for any vR3.

  1. Does this transformation come from multiplication by a matrix?
  2. If so, what is the matrix?
Solution
  1. First, we have just seen that T(v)=proju(v) is linear. Therefore by Theorem 5.2.1, we can find a matrix A such that T(x)=Ax.
  2. The columns of the matrix for T are defined above as T(ei). It follows that T(ei)=proju(ei) gives the ith column of the desired matrix. Therefore, we need to find proju(ei)=(eiuuu)u For the given vector u, this implies the columns of the desired matrix are 114[123],214[123],314[123] which you can verify. Hence the matrix of T is 114[123246369]

This page titled 5.2: The Matrix of a Linear Transformation I is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

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