Julia Goedecke

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 \(\mathbb{R}^n\).

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

Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Introductory question where the numbers come out quite nice with not much dividing. Set-up is meant for formative assessment. Adapated from a question copied from Newcastle.

Given vector $\boldsymbol{v}$  find the norm. Since putting in square roots is tricky, actually input the square norm, so it's an integer.

Decide if matrix sizes match so they can be added.

Given vector $\boldsymbol{v}$  find the norm. Since putting in square roots is tricky, actually input the square norm, so it's an integer.

Simple questions to check comprehension of definition of Euclidean inner product and norm. Meant for formative use.

Simple vector addition and scalar multiplication in \(\mathbb{R}^2\).

Adding and subtracting vectors of random size, including resolving brackets. Advice (i.e. solution) has conditional visibility to show only the correct size.

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.

give the negative of each of two vectors. One always has 5 entries, the other has a random number of entries.

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.

Simple scalar multiplication of a general vector with the important scalars 0, 1, -1. Just the variable name is randomised.

Find the size of a matrix.

Easy true/false questions to check if the meaning of a size of a matrix is understood, in terms of numbers of rows and columns.

Matrix addition, with the added test of whether they understand that only matrices of the same size can be added.

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.

Adding vectors of random size. Advice (i.e. solution) has conditional visibility to show only the correct size.

Adding matrices of random size: two to four rows and two to four columns. Advice (i.e. solution) has conditional visibility to show only the correct size.

Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Set up so that sometimes it has infinitely many solutions (one free variable), sometimes unique solution. Scaffolded so meant for formative. The variable d determines the cases (d=1: unique solution, d-0: infinitely many solutions). The other variables are set up so that no entries become zero for some randomisations but not others.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Not randomized because it's the same as in our workbook. But the variables are made in a way that it should be easy to randomise the size of the matrix, and the to change the formula for the input in not too many places.

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.

Calculate matrix times vector.

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

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