1. Linear independence in \(\mathbb R^3\)
Enter three vectors as columns. The tool row-reduces the matrix \(A=[v_1\ v_2\ v_3]\) and tests whether \(Ax=0\) has only the trivial solution.
Matrix \(A=[v_1\ v_2\ v_3]\)
2. Coordinates in a basis of \(\mathbb R^2\)
A basis is a coordinate system. Enter basis vectors \(b_1,b_2\) and a vector \(v\). The tool solves \(P_{\mathcal B}c=v\).
Basis matrix \(P_{\mathcal B}=[b_1\ b_2]\)
Vector \(v\)
3. Basis from a spanning list
Enter four vectors in \(\mathbb R^3\). The pivot columns of the original matrix form a basis for the span.
4. Rank--nullity for \(T:\mathbb R^4\to\mathbb R^3\)
Let \(T(x)=Ax\). Edit the \(3\times4\) matrix and compute RREF, pivot columns, an image basis, a kernel basis, rank, and nullity.
5. Change of basis in \(\mathbb R^2\)
Input \([v]_{\mathcal B}\). The tool computes \(v=P_{\mathcal B}[v]_{\mathcal B}\), then \([v]_{\mathcal C}=P_{\mathcal C}^{-1}v\).
Basis \(\mathcal B\)
Basis \(\mathcal C\)
Coordinates \([v]_{\mathcal B}\)
6. Dimension counter
Use this to check the rank--nullity identity \(\dim V=\operatorname{rank}(T)+\operatorname{nullity}(T)\).
7. Independent-study practice tasks with answers
Task A. Decide whether \((1,-3,4),(2,-2,5),(3,-1,6)\) are independent.
Answer: They are dependent because \(v_1-2v_2+v_3=0\).
Answer: They are dependent because \(v_1-2v_2+v_3=0\).
Task B. For \(b_1=(1,1)\), \(b_2=(1,-1)\), find \([v]_{\mathcal B}\) for \(v=(4,2)\).
Answer: \([v]_{\mathcal B}=(3,1)\), because \(3b_1+1b_2=(4,2)\).
Answer: \([v]_{\mathcal B}=(3,1)\), because \(3b_1+1b_2=(4,2)\).
Task C. If \(A:\mathbb R^4\to\mathbb R^3\) has rank \(2\), what is the nullity?
Answer: \(4-2=2\).
Answer: \(4-2=2\).
Similar practice. If \(T:\mathbb R^7\to\mathbb R^5\) has rank \(4\), what is \(\dim\ker(T)\)?
Answer: \(7-4=3\).
Answer: \(7-4=3\).
Task D. Ask an AI tool to explain rank--nullity using information preserved/lost. Check whether it correctly identifies image as preserved information and kernel as lost directions.