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Ugur's copy of Find eigenvalues, characteristic polynomial and a normalised eigenvector of a 3x3 matrix
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Question in Ugur's workspace by Ugur Efem and 1 other

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$ .
Characteristic poly, eigenvalues and eigenvectors 3x3, digonailsability (non-randomised)
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Question in Ugur's workspace by Ugur Efem and 1 other

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Ugur's copy of Determine if vectors form a spanning set
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Question in Ugur's workspace by Ugur Efem and 1 other

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.
Combine linearly dependent vectors to get zero
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$ . Find $a,\;b,\;c,\;d$ .
Row reducing a matrix and finding its rank and nullity-MA2223
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.
Find characteristic polynomial, eigenvalues and eigenvectors of a 2x2 matrix
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Question in Julia Goedecke's contributions by Julia Goedecke and 1 other

Example of an explore mode question. Student is given a 2x2 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Characteristic poly, eigenvalues and eigenvectors 3x3 non-randomised
Ready to use
Question in Julia Goedecke's contributions by Julia Goedecke

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Kernel and image of matrix transformation
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Question in Julia Goedecke's contributions by Julia Goedecke and 1 other

Student finds a basis for kernel and image of a matrix transformation. Any basis can be entered; there is a custom marking algorithm which checks if it is a correct basis.

There are options to adjust this question fairly easily, for example to get different variants for practice and for a test, by changing the options in the "pivot columns" in the variables. You should be careful to think about and test your pivot options, as some are easier or harder than others, and some don't work very well.
Gaussian elimination calculator
Ready to use
Question in Julia Goedecke's contributions by Julia Goedecke and 1 other

Educational calculation tool rather than "question".

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".

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.
Inverse algorithm calculator
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Question in Julia Goedecke's contributions by Julia Goedecke and 1 other

Educational calculation tool rather than "question".

This allows the student to input a square matrix (max rows 5). 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 matrix and the identity matrix (or what it has got to). This question has no marks and no feedback as it's just meant as a "calculator". It has some checks in so students know when they are not entering a square matrix or a valid row number etc.

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.
Invertibility via eigenvalues
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Question in Julia Goedecke's contributions by Julia Goedecke

Example of an explore mode question. Student is given a 2x2 matrix with eigenvalues and eigenvectors, and is asked to decide if the matrix is invertible. If yes, second and third parts are offered where the student should give the eigenvalues and eigenvectors of the inverse matrix.

Assessed: remembering link between 0 eigenvalue and invertibility. Remembering link between eigenvalues and eigenvectors of matrix and its inverse.

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Are given matrices in (reduced) row echelon form?
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.
José's copy of Matrix manipulation 2
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Question in Jos's workspace by Jos Klenner

Exercises in multiplying matrices.
Matrices: Manipulation
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Elementary Exercises in multiplying matrices.
Gauss elimination to solve a system of linear equations.
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 2 others

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.
Determine if vectors form a spanning set
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.
Finding bases for the null space and row space of a matrix
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.
Find a basis given a spanning set of vectors
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a basis.
Do three given vectors form a spanning set and are they linearly independent?
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$ .
Invert a 3x3 matrix using row operations
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$ .
Find eigenvalues, characteristic polynomial and a normalised eigenvector of a 3x3 matrix
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$ .
Extract common factors of polynomials
Has some problems
Question in Transition to university by Bradley Bush and 2 others

Factorise polynomials by identifying common factors. The first expression has a constant common factor; the rest have common factors involving variables.
delete syntax checkinbg bryan's copy of MA4100 4.4 Gaussian Elimination
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Question in bryan's workspace by bryan pearce

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.
Reduced Row Echelon Form; spanning a subspace
Draft
Question in Matrices Questions by Lois Rollings

Given two ordered sets of vectors $S,\;T$ in $\mathbb{R^5}$ find the reduced echelon form of the matrices given by $S$ and $T$ and hence determine whether or not they span the same subspace.
MA4100 4.4 Gaussian Elimination
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Question in MA4100 by James Denholm-Price

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.
Inverse Matrix
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Question in Christian's workspace by Christian Lawson-Perfect

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$ .
Ricardo's copy of Inverse Matrix
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Question in Ricardo's workspace by Ricardo Monge

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$ .
Matrix manipulation 2
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Question in Bill's workspace by Bill Foster and 1 other

Exercises in multiplying matrices.
Katie's copy of Linear combinations of 2 x 2 matrices
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Question in Katie's workspace by Katie Dexter

Linear combinations of $2 \times 2$ matrices. Three examples.
Linear combinations of 2 x 2 matrices
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Question in Bill's workspace by Bill Foster and 1 other

Linear combinations of $2 \times 2$ matrices. Three examples.

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