Write \(3\mathbf i-2\mathbf j+5\mathbf k\) as coordinates in the standard basis.

Express \((4,-1)\) as a combination of \(\mathbf i\) and \(\mathbf j\).

For basis \(\mathbf b_1=(1,1)\), \(\mathbf b_2=(1,-1)\), find coordinates of \(\mathbf v=(6,2)\).

Reconstruct a vector with basis coordinates \([3,-1]_B\) for \(B=((2,0),(1,1))\).

Find the magnitude of \(2\mathbf i-3\mathbf j+6\mathbf k\).

Find the standard coordinates of \(2\mathbf b_1+5\mathbf b_2\) where \(\mathbf b_1=(1,0,1)\), \(\mathbf b_2=(0,1,-1)\).

For \(B=((1,2),(3,1))\), find \([\mathbf v]_B\) when \(\mathbf v=(7,8)\).

Show that \((1,2)\) and \((2,4)\) do not form a basis of \(\mathbb R^2\).

Find coordinates of \(\mathbf v=(3,4,5)\) in basis \(\mathbf e_1=(1,0,0)\), \(\mathbf e_2=(0,1,0)\), \(\mathbf e_3=(0,0,1)\).

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

Decompose \((5,5)\) into components parallel to \((1,1)\) and \((1,-1)\).

Convert polar components \(6\) at angle \(60^\circ\) into standard basis components.

For \(B=((1,0),(1,1))\), convert standard vector \((2,5)\) to \(B\)-coordinates.

For \(B=((2,0),(0,3))\), find the standard vector with coordinates \([4,-2]_B\).

Check whether \((1,0,1),(0,1,1),(1,1,2)\) is a basis of \(\mathbb R^3\).

Find the coordinate vector of \(\mathbf v=(0,7)\) in basis \(B=((1,1),(-1,1))\).

Find the orthonormal-basis coordinates of \(\mathbf v=(2,3)\) in basis \(\mathbf u_1=(1/\sqrt2,1/\sqrt2)\), \(\mathbf u_2=(1/\sqrt2,-1/\sqrt2)\).

A vector has coordinates \([1,2,3]_B\) with \(B=((1,0,0),(1,1,0),(1,1,1))\). Find standard coordinates, then recover the \(B\)-coordinates from your result.

A student claims \([2,c]_{((1,2),(3,4))}\) represents \((11,16)\). Find \(c\) from the \(x\)-component and use the \(y\)-component to check whether the claim is consistent.

Find the change-of-basis coordinates of \((1,0)\) and \((0,1)\) in basis \(((1,1),(1,-1))\), then use them to express \((3,-2)\) in that basis.