Vector Spaces and Linear Transformations: math test, College – download the PDF

1 A subspace of R^3 / 4 pts

Let \(W=\{(x,y,z)\in\mathbb{R}^3 : 2x-y+z=0\}\).

1. Show that \(W\) is a subspace of \(\mathbb{R}^3\).
2. Find a basis of \(W\) and its dimension.

2 Dependence / 3 pts

Are \((1,0,2)\), \((2,1,1)\), \((0,-1,3)\) linearly independent? If not, give a dependence relation.

3 Null space and column space / 4 pts

Let \(M=\begin{pmatrix}1&-1&2&0\\2&-2&5&1\\1&-1&3&1\end{pmatrix}\). Find the reduced echelon form, a basis of \(\mathrm{Nul}(M)\), a basis of \(\mathrm{Col}(M)\), and check rank-nullity.

4 A linear map and its kernel / 3 pts

Let \(T(x,y,z)=(x+y-z,\,2y+z)\). Give the matrix of \(T\), a basis of \(\ker T\), and decide whether \(T\) is onto \(\mathbb{R}^2\).

5 Change of basis / 4 pts

Let \(B=\{(3,1),(1,1)\}\) and \(v=(7,5)\).

1. Write \(P\) and \(P^{-1}\).
2. Find \([v]_B\).
3. Let \(T(x,y)=(2x,\,y)\). Find \([T]_B\).

6 True or false? / 2 pts

True or false? Justify. (a) The union of two subspaces of \(\mathbb{R}^2\) is always a subspace. (b) Any four vectors in \(\mathbb{R}^3\) are linearly dependent.