What two properties must a list have to be a basis for a vector space \(V\)?

Why is \(\{(1,0),(0,1)\}\) called the standard basis of \(\mathbb R^2\)?

Can \(\{(1,0),(2,0)\}\) be a basis for \(\mathbb R^2\)?

Is a single non-zero vector a basis for the line it spans?

Check whether \(B=\{(1,1),(1,-1)\}\) is a basis for \(\mathbb R^2\).

Check whether \(B=\{(1,2),(2,4)\}\) is a basis for \(\mathbb R^2\).

Find a basis for \(W=\{(x,y,0):x,y\in\mathbb R\}\).

Find a basis for the line \(L=\{t(2,-3,1):t\in\mathbb R\}\).

Show that \(B=\{(1,0,1),(0,1,1),(0,0,1)\}\) is a basis for \(\mathbb R^3\).

Decide whether \(\{1,x,x^2\}\) is a basis for \(P_2\), the polynomials of degree at most \(2\).

Explain why a basis gives unique coordinates.

Why is a spanning list with a redundant vector not a basis? Use \((1,0),(0,1),(1,1)\).

For which \(k\) is \(\{(1,1),(1,k)\}\) a basis for \(\mathbb R^2\)?

For which \(a\) is \(\{(1,0,a),(0,1,a),(1,1,2a)\}\) a basis for \(\mathbb R^3\)?

For which \(t\) is \(B_t=\{(1,0,0),(0,1,0),(1,1,t)\}\) a basis for \(\mathbb R^3\)?

For which \(a\) is \(\{1+x,1+ax\}\) a basis for \(P_1\)?

For which \(b\) do \((1,0,b),(0,1,b),(1,1,1)\) form a basis for \(\mathbb R^3\)?

A student says every spanning set is a basis. Diagnose the error.

Prove that if a list is a basis, no vector in the list can be removed without losing spanning.

Prove that if a list is a basis, adding any vector from the same space makes the enlarged list dependent.