What does \([\mathbf v]_B\) mean?

Why must a basis be ordered before coordinates are written?

Find \([(3,4)]_B\) for the standard basis \(B=((1,0),(0,1))\).

If \(B=((0,1),(1,0))\), find \([(3,4)]_B\).

Find coordinates of \((5,1)\) in \(B=((1,1),(1,-1))\).

Check that \([(5,1)]_{((1,1),(1,-1))}=(3,2)\).

Find \([(2,3)]_B\) for \(B=((1,0),(1,1))\).

Find \([(1,2,3)]_B\) for \(B=((1,0,0),(0,1,0),(1,1,1))\).

Find the vector with coordinates \((2,-1)\) in \(B=((1,2),(3,1))\).

For \(B=((1,1),(2,1))\), find \([(4,3)]_B\).

Explain why coordinates in a fixed basis are unique.

A vector has coordinates \((1,2)\) in \(B=((1,0),(1,1))\). Find its standard coordinates and explain the basis dependence.

Find \([(x,y)]_B\) for \(B=((1,1),(1,-1))\).

Find coordinates of \((a,b,c)\) in \(B=((1,0,0),(1,1,0),(1,1,1))\).

For which \(k\) does \(B_k=((1,1),(1,k))\) allow unique coordinates for every vector in \(\mathbb R^2\)?

For which \(t\) is \([(1,2,3)]_{B_t}\) defined for every target vector when \(B_t=((1,0,0),(0,1,0),(1,1,t))\)?

For \(B=((1,0),(1,k))\), find \([(x,y)]_B\) when possible and state the condition on \(k\).

A student writes \([(3,4)]_B=(3,4)\) for every basis \(B\). Give a counterexample.

Prove that changing the order of a basis changes the order of coordinates.

Explain why coordinates describe the same vector, not a different physical displacement.