Problems

Bases Coordinates and Subspaces

For these problems, you should use SymPy to find row reduced echelon matrices. Once you write a matrix \(A\), you can simply write the matrix \(R\) indicating that it is the row reduced one. You can use symbols like this: \(A\xrightarrow{\text{RREF}}R\).

Problem 1

Let \(\mathcal{B} = \left\{\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}\right\}\) be a basis for \(\mathbb{R}^3\).

  1. Find the coordinates of \(\mathbf{v} = (3,2,4)\) with respect to \(\mathcal{B}\). That is, find scalars \(c_1, c_2, c_3\) such that:

\[\mathbf{v} = c_1(1,1,0) + c_2(-1,1,0) + c_3(0,0,2)\]

  1. Find the coordinates of the following vectors with respect to \(\mathcal{B}\) simultaneously. You can solve three systems of equations, but if you pay attention, you can solve one system that gives you the coordinates of the three vectors at once.
  • \(\mathbf{v}_1 = (1,0,1)\)
  • \(\mathbf{v}_2 = (2,2,0)\)
  • \(\mathbf{v}_3 = (-1,1,3)\)

Problem 2

Given the basis \(\mathcal{B} = \{(1,1,1), (1,1,-1), (1,-2,0)\}\) for \(\mathbb{R}^3\):

  1. Find the coordinates of the standard basis vectors with respect to \(\mathcal{B}\):
  • \(\mathbf{e}_1 = (1,0,0)\)
  • \(\mathbf{e}_2 = (0,1,0)\)
  • \(\mathbf{e}_3 = (0,0,1)\)

Problem 3

Let \(\mathcal{B} = \{(2,0,1), (0,1,1), (1,1,0)\}\) be a basis for \(\mathbb{R}^3\).

  1. Find the vector \(\mathbf{v}\) whose coordinates with respect to \(\mathcal{B}\): \([v]_\mathcal{B}=(2,-1,3)\).

  2. If a vector has coordinates \((1,1,1)\) with respect to \(\mathcal{B}\), what is this vector in standard coordinates?

Problem 4

Let \(S_1 = \left\{ \begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}0\\2\\1\end{bmatrix}, \begin{bmatrix}1\\1\\2\end{bmatrix} \right\}\) and \(S_2 = \left\{ \begin{bmatrix}2\\1\\0\end{bmatrix}, \begin{bmatrix}1\\0\\2\end{bmatrix}, \begin{bmatrix}3\\1\\1\end{bmatrix} \right\}\) be bases of \(\mathbb{R}^3\).

  1. If \([v]_{S_1} = (2,-1,3)\), find \([v]_{S_2}\).
  2. If \([v]_{S_1} = (1,2,0)\), find \([v]_{S_2}\).
  3. If \([v]_{S_1} = (0,0,1)\), find \([v]_{S_2}\).

Problem 5

Let \(S_1 = \left\{ \begin{bmatrix}2\\1\\1\end{bmatrix}, \begin{bmatrix}1\\2\\0\end{bmatrix}, \begin{bmatrix}0\\1\\3\end{bmatrix} \right\}\) and \(S_2 = \left\{ \begin{bmatrix}1\\1\\1\end{bmatrix}, \begin{bmatrix}2\\0\\1\end{bmatrix}, \begin{bmatrix}1\\2\\2\end{bmatrix} \right\}\) be bases of \(\mathbb{R}^3\).

  1. If \([w]_{S_1} = (1,-1,2)\), find \([w]_{S_2}\).
  2. If \([w]_{S_1} = (3,-0,0)\), find \([w]_{S_2}\).
  3. If \([w]_{S_1} = (2,2,1)\), find \([w]_{S_2}\).

Problem 6

For the following matrices

  1. Find a basis for \(\text{col}(A)\)
  2. Find a basis for \(\text{nul}(A)\)
  3. Write the columns of \(A\) that you didn’t include in the basis in terms of the ones you kept

  1. \(A=\begin{bmatrix}1&4&-1&3\\1&5&0&2\\0&3&3&1 \end{bmatrix}\)

  2. \(A=\begin{bmatrix} 5 & 10 & 12 & 14 & 11 & 3\\ -4 & -8 & -5 & -2 & -27 & -16\\ 2 & 4 & 6 & 8 & 0 & -2 \\1 & 2 & 8 & 14 & -19 & -15 \end{bmatrix}\)

Problem 7

For each of the following sets,

  1. write the set as the span of some vectors,
  2. express the set as the column space of a matrix \(A\),
  3. find a basis of the subspace and find the dimension.

Notice that it follows from (1) or (2) that the set is a subspace

  1. The set of all vectors of the form \[ \left\{ \begin{bmatrix} x + 2y \\ y - z \\ 2x - z \end{bmatrix} : x,y,z \in \mathbb{R} \right\} \]

  2. The set of all vectors of the form \[ \left\{ \begin{bmatrix} -3s + 6t - u + v - 7w \\ s - 2t + 2u + 3v - w \\ 2s - 4t + 5u + 8v - 4w \end{bmatrix} : s,t,u,v,w \in \mathbb{R} \right\} \]

  3. The set of all vectors of the form \[ \left\{ \begin{bmatrix} -2s - 5t + 8u - 17w \\ s + 3t - 5u + v + 5w \\ 3s + 11t - 19u + 7v + w \\ s + 7t - 13u + 5v - 3w \end{bmatrix} : s,t,u,v,w \in \mathbb{R} \right\} \]