Within many proofs in mathematics, it is important to be able to demonstrate when two mathematical statements are logically equivalent to each othe
Within many proofs in mathematics, it is important to be able to demonstrate when two mathematical statements are logically equivalent to each other. There are a number of statements that are logically equivalent to the following:
The n × n matrix A is invertible.
Ten of these equivalent statements are given below:
• A is an invertible matrix.
• A is row equivalent to the n × n identity matrix.
• A has n pivot positions.
• The equation Ax = 0 has only the trivial solution.
• The equation Ax = b has at least one solution for each b in Rn.
• The columns of A span Rn.
• The linear transformation x → Ax maps Rn onto Rn.
• There is an n × n matrix C such that CA = I.
• There is an n × n matrix D such that AD = I.
• The columns of A form a basis of Rn.
A. Provide a definition for logical equivalence.
Note: This definition should be structured so that it can be employed in the parts of the task that follow.
B. Provide an interpretation for the given statement: The n × n matrix A is invertible.
1. Explain what this statement means to you.
C. Write a
ief essay (suggested length of 1–2 pages) in which you do the following:
1. Justify that the ten statements are logically equivalent to the statement “The n × n matrix A is invertible.”
The justification does not need to constitute a formal proof, and you may restrict your arguments to R2 (n = 2). For example, you can justify that matrix A has a non-zero determinant by noting that a zero determinant would introduce division by zero in the formula for finding A-1. (Do not use this justification, however, since it was provided as an example.)
2. Explain how each step in your justifications relates to your answers to parts A and B.