Project: matrix transformations

Section 2.4 Project: matrix transformations

This worksheet deals with matrix transformations, and in particular, kernel and image. The goal is to understand these important subspaces in a familiar context.

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Let \(A\) be an \(m\times n\) matrix. We can use \(A\) to define a transformation \(T_A:\R^n\to \R^m\) given by \(T_A(\xx)=A\xx\text{,}\) where we view \(\xx\) as an \(n\times 1\) column vector.

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The kernel of \(T_A\) is the set of vectors \(\xx\) such that \(T_A(\xx)=\zer\text{.}\) That is, \(\ker T_A\) is the set of solutions to the homogeneous system \(A\xx = \zer\text{.}\)

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The image of \(T_A\) (also known as the range of \(T_A\)) is the set of vectors \(\yy\in \R^m\) such that \(\yy = A\xx\) for some \(\xx\in\R^n\text{.}\) In other words, \(\im(T_A)\) is the set of vectors \(\yy\) for which the non-homogeneous system \(A\xx=\yy\) is consistent.

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Because \(T_A\) is a linear transformation, we can compute it as long as we’re given its values on a basis. If \(\{\vv_1, \vv_2,\ldots, \vv_n\}\) is a basis for \(\R^n\text{,}\) then for any \(\xx\in\R^n\) there exist unique scalars \(c_1,c_2,\ldots, c_n\) such that

\begin{equation*} \xx = c_1\vv_1+c_2\vv_2+\cdots + c_n\vv_n\text{,} \end{equation*}

and since \(T_A\) is linear, we have

\begin{equation*} T_A(\xx)=c_1T_A(\vv_1)+c_2T_A(\vv_2)+\cdots +c_nT_A(\vv_n)\text{.} \end{equation*}

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The main challenge, computationally speaking, is that if our basis is not the standard basis, some effort will be required to write \(\xx\) in terms of the given basis.

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Exercise 2.4.1.

Confirm that

\begin{equation*} B=\left\{\begin{bmatrix}1\\0\\2\\3\end{bmatrix},\begin{bmatrix}4\\2\\0\\-3\end{bmatrix}, \begin{bmatrix}0\\4\\-3\\2\end{bmatrix}, \begin{bmatrix}3\\5\\-2\\1\end{bmatrix}\right\} \end{equation*}

is a basis for \(\R^4\text{.}\)

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To assist with solving this problem, a code cell is provided below. Recall that you can enter the matrix \(\begin{bmatrix}a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i\end{bmatrix}\) as Matrix([[a,b,c],[d,e,f],[g,h,i]]) or as Matrix(3,3,[a,b,c,d,e,f,g,h,i]).

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The reduced row-echeleon form of A is given by A.rref(). The product of matrices A and B is simply A*B. The inverse of a matrix A can be found using A.inv() or simply A**(-1).

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One note of caution: in the HTML worksheet, if you don’t import sympy as your first line of code, you’ll instead use Sage syntax. Sage uses A.inverse() instead of A.inv().

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In a Jupyter notebook, remember you can generate additional code cells by clicking on the + button.

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You can also use the cell below to write down any necessary explanation.

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Exercise 2.4.2.

Write each of the standard basis vectors in terms of this basis.

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Suggestion: in each case, this can be done by solving a matrix equation, using the inverse of an appropriate matrix.

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A linear transformation \(T:\R^4\to \R^4\) is now defined as follows:

\begin{equation*} T\left(\begin{bmatrix}1\\0\\2\\3\end{bmatrix}\right)=\begin{bmatrix}3\\0\\2\\-1\end{bmatrix}, T\left(\begin{bmatrix}4\\2\\0\\-3\end{bmatrix}\right)=\begin{bmatrix}1\\2\\0\\5\end{bmatrix}, T\left(\begin{bmatrix}0\\4\\-3\\2\end{bmatrix}\right)=\begin{bmatrix}4\\2\\2\\4\end{bmatrix}, T\left(\begin{bmatrix}3\\5\\-2\\1\end{bmatrix}\right)=\begin{bmatrix}2\\4\\0\\10\end{bmatrix}\text{.} \end{equation*}

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Let \(\{\vece_1,\vece_2,\vece_3, \vece_4\}\) denote the standard basis for \(\R^4\text{.}\)

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Exercise 2.4.3.

Determine \(T(\vece_i)\) for \(i=1,2,3,4\text{,}\) and in so doing, determine the matrix \(A\) such that \(T=T_A\text{.}\)

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Exercise 2.4.4.

Let \(M\) be the matrix whose columns are given by the values of \(T\) on the basis \(B\text{.}\) (This would be the matrix of \(T\) if \(B\) was actually the standard basis.) Let \(N\) be the matrix whose inverse you used to solve part (b). Can you find a way to combine these matrices to obtain the matrix \(A\text{?}\) If so, explain why your result makes sense.

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Next we will compute the kernel and image of the transformation from the previous exercises. Recall that when solving a homogeneous system \(A\xx = \zer\text{,}\) we find the RREF of \(A\text{,}\) and any variables whose columns do not contain a leading 1 are assigned as parameters. We then express the general solution \(x\) in terms of those parameters.

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The image of a matrix transformation \(T_A\) is also known as the column space of \(A\text{,}\) because the range of \(T_A\) is precisely the span of the columns of \(A\text{.}\) The RREF of \(A\) tells us which columns to keep: the columns of \(A\) that correspond to the columns in the RREF of \(A\) with a leading 1.

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Let \(T\) be the linear transformation given in the previous exercises.

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Exercise 2.4.5.

Determine the kernel of \(T\text{.}\)

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Exercise 2.4.6.

Determine the image of \(T\text{.}\)

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Exercise 2.4.7.

The Dimension Theorem states that for a linear transformation \(T:V\to W\text{,}\) where \(V\) is finite-dimensional,

\begin{equation*} \dim V = \dim\ker(T)+ \dim\im(T)\text{.} \end{equation*}

Confirm that your results on this worksheet agree with this theorem.

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You have attempted of activities on this page.

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