Lab 3 Interactive: Linear Spaces, Subspaces, and Linear Maps

Explore span, subspaces, direct sums, kernels, images, quotient dimensions, and Kronecker products. The goal is to make abstract vector-space language computational and visual.

1. Subspace test in the plane

A line \(ax+by=c\) is a subspace of \(\mathbb{R}^2\) exactly when it passes through the origin, i.e. \(c=0\).

2. Geometry of a line through the origin

3. Span and membership in R³

Enter two generators \(u,v\) and a target vector \(w\). The tool solves \(su+tv=w\).

u

v

w

4. Direct sum in R²

Two lines through the origin give \(\mathbb{R}^2=U\oplus W\) if they are different lines.

5. Kernel and image of a linear transformation

Let \(T:\mathbb{R}^4\to\mathbb{R}^3\) be given by \(T(x)=Ax\). Edit the matrix and compute RREF, pivot columns, image basis, kernel basis, rank, and nullity.

6. Quotient dimension

For finite-dimensional spaces, \(\dim(V/N)=\dim(V)-\dim(N)\).

7. Kronecker product

The Kronecker product is a matrix model for tensor-product computations.

A

B

8. Independent-study practice tasks with answers

Task A. Decide whether \(\{(x,y):2x-y=0\}\) is a subspace.
Answer: Yes. It contains the origin and is the kernel of the linear functional \(T(x,y)=2x-y\).
Task B. Decide whether \(\{(x,y):2x-y=1\}\) is a subspace.
Answer: No. It does not contain \((0,0)\). It is an affine line, not a linear subspace.
Task C. For \(u=(1,0,2)\), \(v=(0,1,-1)\), decide whether \(w=(3,4,2)\) belongs to \(\operatorname{span}(u,v)\).
Answer: Solve \(su+tv=(s,t,2s-t)=(3,4,2)\). Then \(s=3\), \(t=4\), and \(2s-t=2\). So yes.
Similar practice. Try \(w=(3,4,1)\) with the same \(u,v\).
Answer: No, because \(2s-t=2\) for \(s=3,t=4\), not \(1\).
Task D. If \(T:\mathbb{R}^4\to\mathbb{R}^3\) has rank \(2\), what is \(\dim\ker(T)\)?
Answer: By rank-nullity, \(\dim\ker(T)=4-2=2\).
Task E. If \(N\) is a 2-dimensional subspace of \(\mathbb{R}^4\), what is \(\dim(\mathbb{R}^4/N)\)?
Answer: \(4-2=2\).

9. Workspace