Linear Algebra

Let's see how Wolfram|Alpha can handle or help handle typical problems in a linear algebra course.

# Vectors

W|A Query: {2, -5, 4}

Result: Wolfram|Alpha sums the entries in the vector (to get 1), finds the average of the components of the vector (1/3 in this case), computes the length of the vector (about 6.7, given as a decimal approximation and exact number involving a square root), and normalizes the vector.

Implications: Having a tool to quickly determine the length of or normalize vector should be handy for students, particularly in solving complex problems that require many such calculations. It's worth noting that there are no "Show Steps" options for these computations, so they don't show students how to perform these computations, for better or for worse.

As usual, Wolfram|Alpha provide more information than is requested. In this case, seeing the length of a vector along with its normalization might help some students see that there is a connection between these two computations, even if Wolfram|Alpha's results don't really shed any light on what that connection is.

# Matrices

W|A Query: {{2,0},{1,3}}

Result: Wolfram|Alpha provides lots of information about a matrix: its determinant, its trace, its eigenvalues and corresponding eigenvectors, its condition number, and its inverse. Perhaps surprisingly, W|A doesn't provide an echelon form of the matrix. All of the results provided, however, can be expressed in approximate or exact form. No "Show Steps" options are provided, however.

W|A also specifies the dimensions of the matrix (which could come in handy if it has been cut-and-pasted from another application) and provides a "matrix plot" in which the values of the matrix entries are represented by colors. See below. Implications: Again, W|A serves as a handy computational aid, but doesn't shed much light on the associated concepts—or even the calculations themselves, given the lack of "Show Steps." Seeing all these computations at once isn't likely to help students who don't already know the connections among them.

## Row Reduce a Matrix

W|A Query: rref[{{2,0},{1,3}}

Result: Wolfram|Alpha provides the reduced row echelon form of the matrix, which in this case is the identity matrix. There is no "Show Steps" option, and no additional information is provided. Note that W|A provides the result in list form, not in matrix notation. To see the result in matrix notation, use the following command: matrixform[rref[ {{0,1,4,-5},{1,3,5,-2},{3,7,7,6}}] ]

Limitations: Wolfram|Alpha appears to have an input limit of 200 characters. This limits the usefulness of Wolfram|Alpha in row-reducing large matrices and matrices with real-world data (which often has several significant figures). For instance, the following command maxxes out Wolfram|Alpha's input limit:

The matrix involved isn't very large, only 5 x 6:

(1)
\begin{align} \left [ \begin{matrix} -0.5893 & 0.0352 & 0.05 & 0.0587 & 0.1093 & 0\\ .0582 & -.5882 & .1241 & .0922 & .2674 & 0\\ .0665 & .1532 & -.799 & .3151 & .4856 & 0\\ .0742 & .1021 & .0773 & -.711 & .1289 & 0\\ .3904 & .2977 & .5476 & .2451 & -.9912 & 0 \end{matrix} \right ] \end{align}

However, the entries have several digits each (for the most part), resulting in the command hitting Wolfram|Alpha's limit.

Wolfram|Alpha apparently has a size limit, too. For example, Wolfram|Alpha can't handle the following command:

The matrix involved, however, is only 5 x 8:

(2)
\begin{align} \left [ \begin{matrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 800\\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 750\\ 0 & -1 & 1 & 0 & 0 & 1 & 0 & 200\\ 1 & -1 & 0 & 1 & 0 & 0 & 0 & 200\\ 0 & 0 & 0 & 1 & 0 & 1 & -1 & 600 \end{matrix} \right ] \end{align}

Omitting one of the rows to produce a 4 x 8 matrix works fine, however.

## Matrix-Vector Multiplication

Example: Multiply the matrix

(3)
\begin{align} \left [ \begin{matrix} .5 & .4 & .6\\ .2 & .2 & .3\\ .3 & .4 & .1 \end{matrix} \right ] \end{align}

by the vector

(4)
\begin{align} \left [ \begin{matrix} 500\\200\\300 \end{matrix} \right ]. \end{align}

Results: Wolfram|Alpha performs the operation as expected, yielding the result {510, 230, 260}. Also provided is the sum and average of the components of this vector, as well as its length, normalization, and spherical coordinates. This is consistent with how Wolfram|Alpha treats vectors.

For some reason (perhaps because the matrix involved is stochastic), Wolfram|Alpha also converts the result into a pie chart: # Linear Systems

## Solve a Matrix Equation

W|A Query: solve {{2,0},{1,3}}.{x1,x2}={3,5}

Results: Wolfram|Alpha provides the solution to this simple matrix equation,

(5)
\begin{align} x_1 = \frac{3}{2}, x_2 = \frac{7}{6}. \end{align}

There is no "Show Steps" option, and no additional information is provided.

page revision: 19, last edited: 01 Oct 2009 01:28