Final answer:
Without additional information about matrix a, it cannot be said with certainty whether there exists a vector z such that ax = z has a unique solution. However, a vector in a two-dimensional space can form a right angle triangle with its components, in line with the Pythagorean theorem.
Step-by-step explanation:
If a student has a matrix a and a vector y such that the equation ax = y does not have a solution, the question is whether there exists a different vector z such that the equation ax = z has a unique solution. The validity of this statement depends on the properties of the matrix a itself. If matrix a is non-singular (invertible), then for any given vector z there would indeed exist a unique solution. However, if the matrix is singular (not invertible), then there will not exist a unique solution for any vector z. Without additional information about matrix a, we cannot definitively answer true or false to this question.
Vector representation often involves the Pythagorean theorem, especially when vectors are described by their components in a coordinate system. For instance, True or False: Every 2-D vector can be expressed as the product of its x and y-components. The answer is False as a vector is represented as the sum of its components, not the product.
Furthermore, True or False: A vector can form the shape of a right angle triangle with its x and y components. The answer is True, as by definition the x and y components of a vector act as the legs of a right-angle triangle, with the vector itself being the hypotenuse, according to the Pythagorean theorem.