Test whether \((1,2)+(3,-5)\) stays in \(\mathbb R^2\).

Test whether \(4(2,-1,0)\) stays in \(\mathbb R^3\).

Decide whether \(S=\{(x,y):y=2x\}\) is closed under addition.

Decide whether \(S=\{(x,y):y=2x+1\}\) contains the zero vector.

Is \(\{(x,y,z):x+y+z=0\}\) a subspace of \(\mathbb R^3\)?

Is \(\{(x,y):x\ge0\}\) a vector subspace of \(\mathbb R^2\)?

Determine whether \((5,1)\) lies in \(\operatorname{span}\{(2,1),(1,-1)\}\).

Express \((4,7)\) as a linear combination of \((1,1)\) and \((1,2)\).

Check whether \(\{(x,y,z):z=0\}\) is closed under scalar multiplication.

Decide whether \((1,1,1)\) is in \(\operatorname{span}\{(1,0,1),(0,1,1)\}\).

Test whether \(S=\{(x,y):xy=0\}\) is closed under addition.

Find a basis for \(\operatorname{span}\{(1,2),(2,4),(0,1)\}\).

Determine whether \(\{(x,y,z):x=0, y=0\}\) is a subspace and identify it geometrically.

Find \(a,b,c\) if \(a(1,0,0)+b(0,1,0)+c(0,0,1)=(3,-2,5)\).

Is \(\{(x,y):x+y=1\}\) a subspace of \(\mathbb R^2\)?

Show that the span of \((1,2,3)\) is closed under addition.

Find all \(t\) such that \((2,t,6)\) lies in \(\operatorname{span}\{(1,2,3)\}\).

Determine whether \(S=\{(x,y,z):x-y+2z=0\}\) is a subspace of \(\mathbb R^3\), and give two independent vectors that span it.

Decide whether \((1,0,0),(0,1,0),(1,1,0)\) spans \(\mathbb R^3\). If not, identify the subspace they do span and give a redundant vector.

Find the dimension of \(\operatorname{span}\{(1,0,1),(0,1,1),(1,1,2)\}\), and determine whether \((2,3,5)\) lies in this span.