Linear Algebra 1st year

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

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.

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.

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.

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

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

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

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.

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

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.

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

Determine for which value of \(t\) two vectors are parallel. In the first part, there is no real number \(t\) to make it work. In the second part, a value can be worked out.

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.

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.

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.

Decide if matrix sizes match so they can be added.

Calculate trace of a matrix. Fixed matrices as the same as in our workbook.

Calculate trace of a matrix.

simple sums of matrices and scalar mult of matrices.

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.