Final answer:
The assertion that UUᵀx = x for all x in Rⁿ, given that U is an n x p matrix with orthonormal columns, is true. UUᵀ acts as an identity matrix or a projection operator, thus preserving the vector x.
Step-by-step explanation:
The question revolves around the property of an n x p matrix U that has orthonormal columns, and whether the product of U with its transpose UT applied to any vector x in Rn yields the original vector x. This can be stated as UUTx = x for all x in Rn. The statement is True.
For a matrix with orthonormal columns, the columns are both orthogonal (perpendicular to each other in n-dimensional space) and normalized (each column vector has a length of 1). The product of U with its transpose UT results in the identity matrix I if U is a square matrix. When U is not square, but the columns are orthonormal, UTU equals I, and UUT acts as a projection operator that projects any vector x in Rn onto the subspace spanned by the columns of U. In both cases, multiplying UUT with x does not change the vector, hence the equation UUTx = x stands correct. Therefore, the correct option in the final part is True.