Instructions

Let \(V\) be a vector space with a basis \(\mathcal{V} = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\), and let \(W\) be a vector space with a basis \(\mathcal{W} = \{\mathbf{w}_1, \ldots, \mathbf{w}_m\}\). For a linear map \(T : V \to W\), the matrix of \(T\) relative to \(\mathcal{V}\) and \(\mathcal{W}\) is the \(m \times n\) matrix \[ A = \begin{bmatrix} [T(\mathbf{v}_1)]_{\mathcal{W}} & [T(\mathbf{v}_2)]_{\mathcal{W}} & \cdots & [T(\mathbf{v}_n)]_{\mathcal{W}} \end{bmatrix}. \] It transforms coordinates: \([T(\mathbf{x})]_{\mathcal{W}} = A[\mathbf{x}]_{\mathcal{V}}\) for every \(\mathbf{x}\) in \(V\).

Each problem is worked in these steps:

  1. Apply \(T\) to each basis vector \(\mathbf{v}_j\) in \(\mathcal{V}\).
  2. Find the coordinates \([T(\mathbf{v}_j)]_{\mathcal{W}}\) of each result relative to \(\mathcal{W}\).
  3. Place the coordinate vectors into the columns of \(A\).
  4. Find the coordinate vector \([\mathbf{x}]_{\mathcal{V}}\) of a vector \(\mathbf{x}\) relative to \(\mathcal{V}\).
  5. Use \(A\) to find the coordinates \([T(\mathbf{x})]_{\mathcal{W}} = A[\mathbf{x}]_{\mathcal{V}}\).

The Practice path chooses the problems for you, in stages from easy to hard. Each stage adds one new idea, such as a nonstandard basis or a map on polynomials, and gives a new random problem of that kind each time. After two problems at a stage with no hints, the page suggests the next stage. To pick the spaces, bases, and kind of map yourself, or to load an example from lecture, choose Choose problems.

Separately, choose a Level for how much of each problem you do, from easiest to hardest:

  1. Assemble \(A\) from given coordinates; the page does steps 1, 2, and 4.
  2. Find coordinates and assemble \(A\); the page does step 1.
  3. Apply \(T\), find coordinates, and assemble \(A\): every step.
  4. Solve the whole problem by hand: work on paper, then enter only \(A\) and the answer to part (b).

Your progress, at the bottom of the page, shows how many problems you've finished at each stage of the practice path, and a log of your recent problems with each one's stage, level, and hints. If you go back to an earlier stage, Continue where you left off returns you to the furthest stage you've reached, at the level you were using there. After three problems at a level with no hints, the page suggests the next level.

Fill in the boxes and click Check (or press Enter); Tab moves between the boxes. For coordinates and the matrix \(A\), a check doesn't say which entries are wrong, since one wrong entry usually means the method went wrong. Click Hint after a wrong check to see what's wrong with your answer. Each step uses the answers of the steps before it, so a step opens once the steps before it are right (or shown). Entries can be integers, decimals, or fractions such as -3, 0.5, or 1/2. Check turns on once every box in the step is filled in (enter 0 for zeros). Stuck? Hint suggests what to do, and clicking it again gives a more specific hint. Show fills in the answer. The page counts the hints you use for each problem; shown answers and wrong checks count as hints too (checking the same wrong answer again doesn't count twice). The problem is saved in this page's address, so you can reload the page or share a link.

Finding coordinates relative to a basis means solving a linear system; to practice row reducing, try Gaussian Elimination Practice. See the definitions of a linear map, and a coordinate vector.

Practice

Level

Problem

Solution

Calculations should be done by hand. Avoid performing calculations in your head. You should have paper or a tablet in front of you as you work on these problems and practice writing out the necessary calculations to find the answers.