Practice question · Numerical answer
The 2x2 matrices with trace 0 (top-left entry plus bottom-right entry equal to zero) form a subspace of the 2x2 matrices. How many of the four entries can still be chosen freely?
Hints
- Write a general matrix [a, b ; c, d] and impose the condition.
- The condition a + d = 0 forces one entry to be determined by another.
Show the answer
3
Why
The condition a + d = 0 means d = -a, so a, b and c are free and d follows: 3 free entries out of 4. One linear condition removes exactly one degree of freedom, the same accounting that will reappear as dimension in Lesson 5.
Practise Subspaces
The app has 7 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Subspaces
- Every subspace must contain the zero vector. Why does that single requirement rule out so many otherwise…
- The subspace test asks for the zero vector first, before checking closure. Why is that the right order to…
- Order the steps of the subspace test applied to a subset W of a vector space V, cheapest check first.
- Select every subset of the plane that DOES contain the zero vector but is still NOT a subspace.
- Complete the consequence of closure under scalar multiplication.