Grasple has multiple check methods using the Computer Algebra System (CAS) which can be used within the editor to indicate when the answer of a student should be marked correct/incorrect. A full list of current options can be found in this article.

This article covers the **same Basis as** check. There are four sections in this article:

- When to use the same Basis as check?
- Example cases
- An example exercise
- How does the same Basis as check work?

There are two similar basis checks: same Orthogonal Basis and same Orthonormal Basis. These have the same logic with in addition the checks on **orthogonality** and **normality** of the vectors in the basis.

## When to use the same Basis as check?

If you want to check whether the student answer is a basis and it describes the same basis as you've provided, use this check. By providing a basis you can easily check whether the student answer describes the same basis without having the write your own logic.

Sometimes you want students to give a orthogonal basis or orthonormal basis. In this case you can use the checks "same Orthogonal Basis" or "same Orthonormal Basis".

## Example cases

Below you can see example cases on when the "same Basis as" is true and when it will return false.

## Example exercise

The following steps describe in short how to do this.

- Create a new exercise with type "
*Math*". - Set the check type of the exercise by changing the answer rule relation to "
*same Basis as*". Read more about answer rules here. - Give a correct basis by typing the column vectors which describe the basis between curly brackets. This basis will be used to test whether the answers given by learners are correct.
- Go to preview mode and test different answers.
- (Optional) add more detailed feedback/explanation to the exercise.

## Steps

#### Step 1: create a new exercise with type "Math"

Build up your exercise as you always do: add a description of the question, add the question itself, provide a correct answer (in the answer box) and (optionally) add more detailed feedback.

See an example below:

#### Step 2: set the check-type in the main answer rule to "same Basis as"

Click on the "edit answers and specific feedback" button and change the relation for the main (i.e. first) answer rule to "**same Basis as**".

#### Step 3: give a correct basis to check answers with

Give a correct basis as the correct answer in the edit view. This basis will be used to check whether the answers given by learners are also correct basises for the subspace.

To create a basis, use the matrix insert menu and comma's to separate the vectors.

**Step 4: test different correct and incorrect answers**

For our sample exercise a correct answer is a basis consistent of v_1 and v_3.

And an incorrect answer for our sample exercise is a list of vectors with all three vectors.

#### Step 5: add more detailed feedback (optional)

Automatic feedback will be shown to student about whether they have the answer correct or incorrect including a correct answer.

However more detailed explanation on when answers are correct help the learners in learning from both their correct as incorrect answers.

You can add general feedback in the green "Detailed solution" box and orange "Question at wrong answer" box. For more **specific feedback** based on the characteristics of the student answer, check out the answer rules options.

## How does the same Basis as check work?

The method checks whether the given basis and the student answer describe same subspace. This is done in three steps:

- Check if the number of vectors are the same
- Check whether the dimension (rank of matrix consistent of the vectors) of the answer is the same
- Check whether they span the same subspace by creating a single matrix with the given basis vectors and the vectors from the student answer and check if the rank has not changed compares to the given basis

#### Orthogonal and orthonormal

For the check method "same Orthogonal Basis as" there is one additional check

- check if vectors in the basis are orthogonal to each other

For the check method "same Orthonormal Basis as" the above check is used and an additional fifth check:

- check if all vectors are normalised (magnitude of 1)

Do you want to know more about this check, whether you should use it for your exercise or any of the other checks? Please let us know via the chat icon in the right bottom of the screen!