ChatGPT in Teaching Linear Algebra: Strides Forward, Steps to Go

3.1.1 Linear Equations

Consider the following system of linear equations:

First-year students recognize easily that System (1) has only a trivial solution. ChatGPT version 3 claimed that the system has infinitely many solutions (Bagno et al., 2023).

A short analysis showed that the software lacks logical reasoning. We emphasize once again that the educator knows from the beginning that the bot has not been built for reasoning, but on a statistical basis. The goal here is to analyze how a student who is not fully aware of this situation will “accept” the bot’s answers, and at which level of critical thinking. While all the performed steps were correct, they were not the most efficient ones, and the software failed to deliver a correct answer (Bagno et al., 2023). As the bot did not use the accurate Gauss algorithm, we requested the use of the exact algorithm and the bot provided us with what we requested.

Three months later, we asked ChatGPT to solve System (1) once again. This time, it solved the system quickly and flawlessly, without explanations. We requested then a detailed description of the process; ChatGPT’s comprehensive explanation is displayed in Figure 2.

Figure 2

Screenshot of a ChatGPT 4 session where the bot was asked to solve a system of linear equations.

Afterward, we broadened the problem, with the following parametric system of linear equations:

We requested the determination of the values of the parameter for which the system has either a unique solution or no solution at all.

ChatGPT wrote the extended matrix of the system and subtracted the first row from the second one and the third one. Its answer is (the bold letters are from the original screen):

1. The system has a  unique solution  for all values of  a   except  a = 1 .

2. The system has  infinite solutions  for  a = − 1 .

3. The system does not have any value of  a   for which there is no solution.

In the typical execution of the Gauss elimination method, one would begin by dividing the second row by 1 − a , under the condition a ≠ 1 ^[4]. The next step would usually involve subtracting the second row from the third one. However, ChatGPT bypassed the initial step and proceeded directly to add − a − 1 1 − a times the second row to the third row.

The exceptional case a = 1 had not been treated. Moreover, the bot has never considered the second possibility, which is a contradiction, for the case a = 1 . The fact that ChatGPT mistreated the case a = 1 caused it not to work properly in recognition of the case where the system had no solutions.

This is partly similar to what happens with a CAS, as analyzed in Dana-Picard (2001): the particular cases are ignored and the software behaves as if the expression in the denominator is always non-zero, which requires a totally different way for the particular cases.

To summarize, it seems that ChatGPT employs methods suited for systems without parameters when addressing systems that include a parameter, leading to a misunderstanding of the complete scenario. The lack of consideration of values for which the determinant vanishes, described in Dana-Picard (2001), appears here.

3.1.2 Eigenvalues Computations

We presented ChatGPT with the straightforward task of determining the eigenvalues of the following matrix:

After numerous attempts in several sessions, our persistence was rewarded with the correct answer. The initial response from the AI included an error in calculating the determinant, as evident in the last row of Figure 3. This error could have been avoided if ChatGPT had opted to apply Laplace expansion to the first column, which contains two zeroes, rather than to the first row.

Figure 3

Algebraic mistakes that ChatGPT made in an attempt to find the eigenvalues of a matrix.

When we observed that the solution provided was incorrect, we pointed this out to ChatGPT without indicating where the error occurred. In response, the bot repeated the calculation, unfortunately making the same algebraic error mentioned earlier. Intriguingly, during this second attempt, ChatGPT committed an additional error, which paradoxically led it to recognize that its calculation could not be correct. The expansion of the parentheses was performed incorrectly, resulting in the characteristic polynomial deviating from its expected third-degree form. This specific step is illustrated in Figure 4. It goes without saying that the outcome obtained from this attempt was incorrect.

Figure 4

ChatGPT’s erroneous step in calculating a determinant.

Note the incorrect language: a polynomial and an equation are two different mathematical objects.

After multiple back-and-forth interactions with the bot, it opted to use the first column rather than the first row for its calculations, which led to the correct computation of the determinant. At this point, we revealed ChatGPT that one of the eigenvalues, specifically λ = 1 , was indeed correct. In subsequent interactions, ChatGPT inserted this piece of information into its computations.

At this point, one might anticipate that ChatGPT would extract the term λ − 1 and treat a second-degree polynomial. Instead, it chose to multiply the polynomial by λ − 1 and, toward the end of the process, to divide by λ − 1 again. The execution of the polynomial division was initially unsuccessful, but it was corrected in a subsequent session after we highlighted the incorrect division.

The lesson one might learn from this episode is twofold:

1. When a user identifies an algebraic or conceptual error in ChatGPT’s response, the most effective approach appears to be explicitly pointing out the specific issue to the bot. A short question such as “solve the system of equations” does not provide the complete answer expected from students, including the special cases. The user should ask separate questions, addressing the different cases. By doing so, the process of arriving at a correct solution can be expedited.

2. ChatGPT does not always employ the most efficient method in solving a given problem. To demonstrate this, we requested the bot to compute the determinant of a 4 × 4 matrix with three zeros in its third column and only two in its first column. Contrary to our expectations, the bot opted for the first column instead of the third one for the Laplace expansion.

As an educational implication, we recommend using ChatGPT in the classroom not as a standalone tutor, but rather as an adjunct to the instructor’s guidance. One effective approach could involve assigning students the task of analyzing and critiquing solutions provided by ChatGPT, encouraging them to identify and discuss any errors or shortcomings in the bot’s responses. After all, critical thinking is one of the already mentioned “four C’s of twenty first-century education” (in Sections 1.2 and 4.2).

3.2.1 Set Recognition

One of the problems that emerged in our first research on ChatGPT (Bagno et al., 2023) was the inability to recognize the membership of a vector in a given vector space. We asked the bot whether the vector ( 1 , 2 , 3 ) belongs to Sp { ( 1 , 0 , 0 ) , ( 1 , 1 , 0 ) } , the subspace spanned by the given vectors in R 3 , or not. It gave us a wrong positive answer. Figure 5, taken from Bagno et al. (2023), is a screen snapshot of ChatGPT 3.0’s answer.

Figure 5

Answer that ChatGPT 3.0 gave to the question: “does a given vector belong to a given vector space?”.

However, an immense improvement has recently been observed. We asked ChatGPT 4.0 the same question, and this time we received a precise answer. The software expressed the probe whether the vector belongs to the given subspace as a vector equation:

which it translated into a system of linear equations, exactly as a human would do. The final output reads as follows:

The equation 0 = 3 is a contradiction and cannot be true. Therefore, the vector ( 1 , 2 , 3 ) does not belong to the span of { ( 1 , 0 , 0 ) , ( 1 , 1 , 0 ) } because there are no scalars a and b that can satisfy these conditions.

We further asked whether the set { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) } spans the entire space R 3 . Again, we received a precise and reasonable answer, performing all the computations from scratch. A human, when asked these two questions in a row, would have used the previous answer about the vector ( 1 , 2 , 3 ) as a counter-example. ChatGPT chose to reinvent the wheel, which can be problematic for a beginner’s learning process.

3.2.2 Identification of Subspaces

As pointed out in Bagno et al. (2023), while ChatGPT generally knows how to explain the necessary conditions for a subset to be a subspace of a vector space, it had a hard time applying it to specific examples. ChatGPT has occasionally failed to recognize that certain sets fulfill these conditions and are subspaces of a given vector space. This situation arose when the bot made an unsuccessful effort to prove that the set A = { ( x , y ) ∣ x , y ≥ 0 } is not a vector subspace of R 2 , which, in itself, is a valid assertion. Furthermore, there was also a lapse in accurately pinpointing the members of this set.

Later, we revisited the question about whether the subset A = { ( x , y ) ∣ x , y ≥ 0 } is a subspace of R 2 . This time, the answer was flawless, as shown in Figure 6.

Figure 6

Flawless response of ChatGPT to our question whether the set { ( x , y ) ∣ x , y ≥ 0 } is a vector space.

Unfortunately, a similar inquiry yielded a much worse response. We asked ChatGPT whether the set { ( x , y ) ∈ R 2 ∣ x y ≥ 0 } is a subspace of R 2 . The answer contained several types of mistakes.

It correctly listed the three conditions for being a subspace and recognized correctly that the set is not closed under addition, but provided an incorrect counter-example. Regarding the closure under scalar multiplication (which does hold in this case), ChatGPT tried without our intervention to bring a counter-example which, as it realized right away, was not helpful, and tried to correct itself by bringing another counter-example, which was also incorrect. It claimed that the set is not a subspace of R 2 , since it is not closed under scalar multiplication. This claim is wrong since the set is closed under scalar multiplication but not under addition (Figure 7). This happened in a single shot, before any further conversation with the bot.

Figure 7

Failed response of ChatGPT to our question whether the set { ( x , y ) ∣ x y ≥ 0 } is a vector space.

3.2.3 A Tricky Question

The following tricky query was posed to ChatGPT: “Is the set S p { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) } linearly independent?” Such a question, when posed to students, is designed to test their understanding of fundamental concepts in linear algebra, in particular the concept of a subspace spanned by a given subset. The span of such a set, being a vector space in itself, is obviously linearly dependent. We anticipated that some students might mix up a set of vectors with its span. Surprisingly, ChatGPT also seemed to have fallen into this common misunderstanding, as illustrated in Figure 8.

Figure 8

ChatGPT’s failed answer to our question about whether the set S p { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) } is linearly independent.

At this step, we chose not to continue with this question.

3.2.4 The Row Space and the Column Space of a Given Matrix

In Bagno et al. (2023), we documented an inquiry with ChatGPT 3.0 about the equality of row space and column space dimensions of given matrices. ChatGPT provided a detailed, informative answer, but inaccurately described the relationship between a matrix’s row space and the column space of its transpose as “the fundamental theorem of linear algebra.”

When asked the same question, ChatGPT 4.0 provided a more accurate and informative answer, including additional detail about the equivalence of row and column spaces in square, full-rank matrices. In particular, it avoided the previous error regarding the fundamental theorem of linear algebra.

Recently, we revisited the same question with ChatGPT 4.0, yet encountered a setback. ChatGPT correctly answered and explained that the row space and the column space of an arbitrary matrix are distinct. However, it suggested that if the matrix is square and orthogonal, these spaces coincide. This part of the response is shown in Figure 9.

Figure 9

ChatGPT’s answer to our question about the equality of row space and column space.

The stipulation of the matrix being orthogonal sounded strange, and we asked ChatGPT about the relevance of orthogonality here. The response can be seen in Figure 10. It seems that ChatGPT is now confused since it claims that a matrix may be orthogonal, without coincidence of the row and column spaces. After we pointed this out to the bot, it apologized (again) and the issue was resolved.

Figure 10

ChatGPT’s apology after we clarified that orthogonality of the matrix has nothing to do with equality of its column and row spaces.

Furthermore, we tried to figure out whether ChatGPT could help us with identifying criteria for a matrix for which its row and column spaces would coincide. This question might be considered as a research problem. We were disappointed to discover that ChatGPT has not learned a lot (Figure 11).

Figure 11

ChatGPT’s disappointing answer to our question about the conditions for equality of column and row spaces of a matrix.

We educate students in our linear algebra course to show critical thinking and recognize the faulty of this response, since for every square and symmetric matrix, the row and column spaces coincide without the matrix being invertible.

3.2.5 Abstract Linear Independence

In Bagno et al. (2023), we asked ChatGPT 3.0 to verify the linear independence of the set { v 1 , v 1 + v 2 , v 1 + v 2 + v 3 } , given that the set { v 1 , v 2 , v 3 } is linearly independent in a given vector space V . Initially, we received a detailed and exact explanation of how to check for linear independence of a set. The explanation was clear, precise, and well articulated, and even the final conclusion was correct. However, the process of solving the needed system of linear equations revealed significant algebraic errors and several steps with incorrect logical reasoning. This observation suggests that ChatGPT tends to use a more visual than logical approach to equation solving. It struggles with methodically working through equations, particularly those involving parameters. In situations lacking a clear algorithmic path, ChatGPT often resorts to improvisation instead of providing an exact solution.

Afterward, we asked ChatGPT again the same question, and this time, we received an excellent answer.

3.2.6 Bases and Dimensions

In Bagno et al. (2023), we asked ChatGPT 3.0 about the dimension of the vector space R n [ x ] of polynomials of degree at most n over R ; we obtained the correct answer: dim R n [ x ] = n + 1 . However, when we asked to apply this information and tell us whether the set of four polynomials { 1 − x , x − x 2 , x 2 − x 3 , x 4 − x 3 } is a basis of the space R 4 [ x ] , we received a detailed and reasoned response, full of systems of equations, and references to definitions and proofs, asserting that the set is indeed a basis of R 4 [ x ] . These explanations included purported proof that this set spans the space. However, it appeared that the proof contained significant logical errors.

We asked a more recent version of the AI (with the same account) the same question about the set { 1 − x , x − x 2 , x 2 − x 3 , x 3 − x 4 } . This time, the answer was correct. The bot noted correctly that since the set we provided has four elements, it cannot be a basis for the given space of polynomials R 4 [ x ] . However, it added some extra information, which was not correct: it claimed that our set did not include polynomials of degree 4.

After some argument with the system, we obtained the complete answer, which can be seen in Figure 12.

Figure 12

Correct and complete answer of ChatGPT to our question about a certain set of polynomials.

3.2.7 Linear Transformations

3.2.7.3 A More Difficult Problem

We tried a more difficult inquiry, namely: how many linear transformations map the line L = { t ( 1 , 1 ) + ( 1 , 0 ) ∣ t ∈ R } to the line M = { t ( 2 , 3 ) + ( 0 , 2 ) ∣ t ∈ R } ? Again, we received an accurate response.

The issue emerged when we asked ChatGPT to construct a linear transformation mapping a line to a point. The bot incorrectly responded that no such linear transformation is possible, reasoning that this would breach the preservation of vector addition and scalar multiplication. This response is illustrated in Figure 13.

Figure 13

No linear transformation mapping a line to a point, really?.

Upon our insistence with a specific example, showing the existence of such a linear transformation, the bot revised its stance, as shown in Figure 14. However, it maintained that our example was atypical within the realm of standard linear transformations, asserting it disrupts the preservation of linear structures, like the convention of lines mapping to lines.

Figure 14

Surreal claims of ChatGPT regarding the existence of a specific linear transformation.

Such an “attitude” is far from fitting a mathematical approach. When exploring a mathematical problem, it is expected to check all the possibilities. They appear generally as a “general” case, and one or several special cases. Nevertheless, when working with technology, it happens that special cases are ignored. Such an issue has been described in Dana-Picard (2001): when asked to solve a parametric system of linear equations, a CAS provided an answer in the general case, but ignored what happens for the special values of the parameter(s). Of course, the same phenomenon appears when inverting a matrix: the given matrix can be invertible in general, and non-invertible for special values of the parameter.

Example 1

Let A = 1 1 k 1 k 1 k 1 1 , where k ∈ R . When asked for an inverse matrix, a computer algebra system answers

A − 1 = − 1 k 2 + k − 2 − 1 k 2 + k − 2 k + 1 k 2 + k − 2 − 1 k 2 + k − 2 k + 1 k 2 + k − 2 − 1 k 2 + k − 2 k + 1 k 2 + k − 2 − 1 k 2 + k − 2 − 1 k 2 + k − 2 .

Discovering a posteriori only that there is a problem (maybe the matrix is non-invertible, but it has not been checked) for k = 1 , − 2 is not an acceptable approach.