Define \(\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_k\}\).

Why does every span contain \(\mathbf0\)?

Describe \(\operatorname{span}\{(2,0)\}\) in \(\mathbb R^2\).

Does \(\{(1,0),(0,1)\}\) span \(\mathbb R^2\)?

Determine whether \((3,4)\) lies in \(\operatorname{span}\{(1,2)\}\).

Determine whether \((3,6)\) lies in \(\operatorname{span}\{(1,2)\}\).

Find coefficients showing that \((5,1)\) is in the span of \((1,1)\) and \((1,-1)\).

Do \((1,0,0)\) and \((0,1,0)\) span \(\mathbb R^3\)?

Show that \((1,0,1)\) and \((0,1,1)\) span the plane \(z=x+y\) in \(\mathbb R^3\).

Do \((1,2)\) and \((2,4)\) span \(\mathbb R^2\)?

Show that \((1,0),(1,1)\) span \(\mathbb R^2\).

Adding a vector already in a span does not change the span. Illustrate using \((1,0),(0,1),(1,1)\).

For which \(k\) do \((1,1)\) and \((1,k)\) span \(\mathbb R^2\)?

Find all \(a\) such that \((1,0,a),(0,1,a),(1,1,2a)\) span \(\mathbb R^3\).

For which \(t\) do \((1,0,0),(0,1,0),(1,1,t)\) span \(\mathbb R^3\)?

For which \(a\) does \(\{(1,a),(a,1),(1,1)\}\) span \(\mathbb R^2\)?

For which \(b\) do \((1,0,b)\), \((0,1,b)\), and \((1,1,1)\) span \(\mathbb R^3\)?

A student says two vectors in \(\mathbb R^3\) can span all of \(\mathbb R^3\) if they are long enough. Explain the error.

Prove that the span of any list of vectors is a subspace.

Prove that if \(T\subseteq S\), then \(\operatorname{span}(T)\subseteq\operatorname{span}(S)\).