Problem 1. (4 × 2 points) For each of the statements below, determine whether it is true or false with justifications (Except Part (iii), if a statement is false, give explicit counterexamples and if it is true, give appropriate arguments). (i) If columns of a square matrix (thought of as column vectors) form a linearly dependent set of vectors, then the same statement is true for the rows. (ii) If S = {V1, V2, V3, V4) is a basis for the vector space R¹, and if W is a subspace of R¹, then some subset of S is a basis for W. (iii) There exists a square matrix A for which det (ATA) = -2023. (If it is true, give an example to show that there exists such a matrix; and if it is false, justify why one cannot find the matrix.) (iv) If V and W are three-dimensional subspaces of R³, then V and W must have a nonzero vector in common (i.e. VnW / {0}).

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Problem 1. (4 × 2 points) For each of the statements below, determine whether it is true or false with justifications (Except Part (iii), if a statement is false, give explicit counterexamples and if it is true, give appropriate arguments). (i) If columns of a square matrix (thought of as column vectors) form a linearl