Linear Algebra 1st year

This allows the student to input a linear system in augmented matrix form (max rows 5, but any number of variables). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the system. This question has no marks and no feedback as it's just meant as a "calculator".

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Adding 2x2 matrices. Very simple question. Marks per correct entry.

In this demo question, you can see either 2 or 3 gaps depending on the variable \(m\), and the marking algorithm doesn't penalise for the empty third gap in cases when it is not shown.

Reason to use it: for vectors or matrices containing only numbers, one can easily use matrix entry to account for a random size of an answer. But this does not work for mathematical expressions. There we have to give each entry of the vector as a separate gap, which then becomes a problem when the size varies. This solves that problem. For this reason I've included two parts: one very simple one that just shows the phenomenon of variable number of gaps, and one which is more like why I  needed it.

Note that to resolve the fact that when \(m=2\), the point for the third gap cannot be earned, I have made it so that the student only gets 0 or all points, when all shown gaps are correctly filled in.

Note the use of Ax[m-1] in the third gap "correct answer" of part b): if you use Ax[2], then it will throw an error when m=2, as then Ax won't have the correct size. So even though the marking algorithm will ignore it, the question would still not work.

Bonus demo if you look in the variables: A way to automatically generate the correct latex code for \(\var{latexAx}\), since it's a variable size. I would usually  need that in the "Advice", i.e. solutions, rather than the question text.

In this demo question, you can see either 2 or 3 gaps depending on the variable \(m\), and the marking algorithm doesn't penalise for the empty third gap in cases when it is not shown.

Reason to use it: for vectors or matrices containing only numbers, one can easily use matrix entry to account for a random size of an answer. But this does not work for mathematical expressions. There we have to give each entry of the vector as a separate gap, which then becomes a problem when the size varies. This solves that problem. For this reason I've included two parts: one very simple one that just shows the phenomenon of variable number of gaps, and one which is more like why I  needed it.

Note that to resolve the fact that when \(m=2\), the point for the third gap cannot be earned, I have made it so that the student only gets 0 or all points, when all shown gaps are correctly filled in.

Note the use of Ax[m-1] in the third gap "correct answer" of part b): if you use Ax[2], then it will throw an error when m=2, as then Ax won't have the correct size. So even though the marking algorithm will ignore it, the question would still not work.

Bonus demo if you look in the variables: A way to automatically generate the correct latex code for \(\var{latexAx}\), since it's a variable size. I would usually  need that in the "Advice", i.e. solutions, rather than the question text.

Marking algorithm that allows NA or any correct counterexample.

Demo of automatically generating latex strings to out put vectors/matrices of variable size and that are calculated by some formula.

Matrix multiplication. Contains a function that will let you print the calculation steps of matrix multiplication, e.g. in the Advice.

checking by size whether two matrices can be multiplied. Student either gives size of resulting product, or NA if matrices can't be multiplied.

First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.

Randomising the number of rows in the matrix, m, makes the marking algorithm for part c) slightly complicated: it checks whether it should include the third gap or not depending on the variable m. For the correct distribution of marks, it is then necessary to do "you only get the marks if all gaps are correct". Otherwise the student would only get 2/3 marks when m=2, so the third gap doesn't appear.

First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.

checking by size whether two matrices can be multiplied. Student either gives size of resulting product, or NA if matrices can't be multiplied.

A combination of tasks: checking which matrix products exist, calculating some of these products, calculating transpose matrices. Comparing product of transpose with transpose of product. Experiencing associativity of matrix multiplication. Not much randomisation, only in which matrix product is computed as second option.

Comprehensive solution written out in Advice.

Calculating with vectors in \(\mathbb{R}^4\), including resolving brackets. The fixed vector size is so that a test is fair to all students.

Abstract linear combinations. "Surreptitious" preview of bases and spanning sets, but not explicitely mentioned. There is no randomisation because it is just an abstract question. For counter-examples, any valid counter-example is accepted.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Randomized size of the matrices and formula.

Calculate matrix times vector.

First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.

Randomising the number of rows in the matrix, m, makes the marking algorithm for part c) slightly complicated: it checks whether it should include the third gap or not depending on the variable m. For the correct distribution of marks, it is then necessary to do "you only get the marks if all gaps are correct". Otherwise the student would only get 2/3 marks when m=2, so the third gap doesn't appear.

This allows the student to input a linear system in augmented matrix form (max rows 5, but any number of variables). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the system. This question has no marks and no feedback as it's just meant as a "calculator".

Calculate matrix times vector.

Checking whether a given set is a plane or not. Depends on whether two vectors are parallel or not. Then checking whether the plane goes through the origin. This is not always obvious from the presentation.

Not randomised because it's the same as in our workbook.

easy vector addition and scalar multiplication, for practice after Section 1 of lectures.

Student can choose one of all possible matrix products from the matrices given. Meant for voluntary extra practice. No extensive solutions: referred to other questions for this.

Matrix multiplication. Has automatically generated "unresolved" matrix product to write in the solution.

A combination of tasks: checking which matrix products exist, calculating some of these products, calculating transpose matrices. Comparing product of transpose with transpose of product. Experiencing associativity of matrix multiplication. Not much randomisation, only in which matrix product is computed as second option.

Comprehensive solution written out in Advice.

Use matrix multiplication to get an equation for \(k\) which is then to be solved.

To understand matrix multiplication in terms of linear combinations of column vectors.

Asking the student to create examples of two matrices which multiply to zero but are not themselves the zero matrix. Then getting the student to think about some features of these examples.

Easy intro questions to be done when the students have seen the "vector space axioms" but not as axioms, just in the context of \(\mathbb{R}^n\).