Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

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 $p,\;q$.

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.

Question requires students to interchange units of Hz with MHz, GHz, THz. Question is not very efficient at present- frequencies spanning many orders of magnitude are generated by variables in a clumsy way. Could be improved by having frequency generated by a 10^((random(1000..4000)/1000) variable instead, for example. 

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.

The student is given a connected graph and must find a minimum spanning tree.

This question asks students to find a spanning tree for simple undirected graphs.

Students are randomly shown one of two networks. They are shown four sub-networks, and asked to identify which one is a minimum spanning tree for the network. Thus, there are 2 versions of this question.

Students are randomly shown one of 3 networks.

They are given 4 sub-networks and asked to identify which one is a spanning tree

This question displays one of 10 graphs. It asks the student to either 

(a) count the vertices, or

(b) count the edges, or

(c) state how many vertices a spanning tree would contain, or

(d) state how many edges a spanning tree would contain, or

(e) state the degree of a selected (randomly chosen) vertex.

The student must make a spanning tree of the complete graph $K_5$. They can tick boxes to include or exclude edges, or toggle them by clicking on a visual representation of the graph.

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.

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.

This question displays one of 10 graphs. It asks the student to either 

(a) count the vertices, or

(b) count the edges, or

(c) state how many vertices a spanning tree would contain, or

(d) state how many edges a spanning tree would contain, or

(e) state the degree of a selected (randomly chosen) vertex.

Students are randomly shown one of two networks. They are shown four sub-networks, and asked to identify which one is a minimum spanning tree for the network. Thus, there are 2 versions of this question.

List all vectors in a spanning set.

(repeated 3 times)

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 $p,\;q$.

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.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

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.