Linear Algebra - Check your knowledge
□ Are you comfortable with the following matrix manipulations?
• Addition, subtraction, and multiplication of matrices, including multiplication
of a matrix by a vector;
• Multiplication of a matrix by a scalar;
• Transposition of matrices.
□ Do you know the definition of linear independence of a set of vectors?
□ Do you know what a vector space is? Do you know how to decide whether a subset of
a given vector space is a subspace (i.e. is itself a vector space)?
□ Do you know what the span of a set of vectors is?
□ Do you know what a basis is? Given a set of vectors, do you know how to decide
whether the vectors in the set form a basis of a given vector space?
□ Do you know how to find the dimension of a vector space? In particular, do you know
how to find the dimensions of the column space, of the row space, and of the null space
of a matrix?
□ How is the rank of a matrix defined?
□ Do you know what the rank theorem says?
□ What is a linear system of equations?
□ Do you know how to decide whether a linear system of equations is consistent?
□ If a system is consistent, how do you know whether it has just one or an infinite
number of solutions?
□ What does it mean for a linear system of equations to be homogeneous? What is the
form of the general solution to a non-homogeneous linear system of equations?
□ Do you know how to find the inverse of a square matrix?
□ Do you know how to expand a determinant with respect to one of its columns or one of
its rows?
□ Do you know what happens to a determinant if you multiply one row by a constant?
What if you multiply 2 rows by the same constant? What if instead of rows, you multiply
some of the columns by a constant?
