Systems of linear equations

Which statement best describes the matrix equation form of a system of linear equations?

• It is only valid for homogeneous systems

• It describes the determinant of a system

• It defines matrix multiplication

If a set of vectors is linearly dependent, this means:

• They form an orthogonal set

• At least one vector can be written as a linear combination of the others

• Their dot products are all zero

• No solution

• Exactly one solution

• Infinitely many solutions

• A determinant equal to zero

Which method can be used to check if a set of vectors is linearly independent?

• Compute the dot product between pairs

• Calculate the determinant of the augmented matrix

• Count the number of solutions of a homogeneous system

• Write the vectors as columns of a matrix and check its rank

What is the augmented matrix of a system?

• A matrix with the solutions in the last column

What condition must be satisfied for a system of linear equations to have no solution?

• All equations are consistent

• Rank(A) = n

• Rank(A) = Rank([A|b])

• Rank(A) < Rank([A|b])

Which of the following is a characteristic of a homogeneous system of linear equations?

• It has no solutions

• It always has a unique solution

• It has at least one non-trivial solution

In the context of linear algebra, what does the rank of a matrix indicate?

• The number of solutions to the system

• The determinant of the matrix

• The number of rows in the matrix

• The maximum number of linearly independent column (row) vectors