## Summary and questions for repetition

I was planning to cover the Moore-Penrose inverse which allows one to solve the equation for any (not necessarily square). Now my feeling is that it would be too much for a standard linear algebra course. This is the most easily accessible sourse.

- Give the definition and example of an orthonormal system. Prove that elements of such a system are linearly independent.
- "To know how a matrix acts on vectors, it is enough to know how it acts on the elements of an orthonormal basis." Explain.
- How can you reveal the elements of a matrix from ?
- A linear mapping from one Euclidean space to another generates a matrix. Prove.
- Prove that an inverse of a linear mapping is linear.
- What is a linear mapping in the one-dimensional case?
- In case of a square matrix, what are the four equivalent conditions for the equation to be good (uniquely solvable for all )?
- Give two equivalent definitions of linear independence.
- List the simple facts about linear dependence that students need to learn first.
- Prove the criterion of linear independence.
- Let the vectors be linearly dependent and consider the regression model Show that here the coefficients cannot be uniquely determined (this is a purely algebraic fact, you don't need to know anything about multiple regression).
- Define a basis. Prove that if is a basis and is decomposed as then the coefficients are unique. Prove further that they are linear functions of
- Prove that the terms in the orthogonal sum of two subspaces have intersection zero.
- Prove dimension additivity.
- Prove that a matrix and its adjoint have the same rank.
- Prove the rank-nullity theorem.
- Prove the upper bound on the matrix rank in terms of the matrix dimensions.
- Vectors are linearly independent if and only if one of them can be expressed as a linear combination of the others.
- What can be said about linear (in)dependence if some vectors are added to or removed from the system of vectors?
- Prove that if the number of vectors in a system is larger than the space dimension, then such a system is linearly dependent.
- Give a list of all properties of rank that you've learned so far.

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