MA351 LINEAR ALGEBRA Ideas and Applications – 20210627y

True-False Questions: Justify your answers. 2.1:A 5 × 27 matrix can have six linearly independent columns 2.4:Suppose that A is an n × n matrix and B is an n × 1 column vector such that the equation AX = B has an infinite number of solutions. Then the columns of A are linearly dependent. 2.5 The functions cos2 x, sin2 x, sin 2x, and cos 2x form a linearly independent set. 107:Let X = [1, 0, 2, 0]t and Y = [1,−1, 0, 2]t . (a) Find a system of two equations in four unknowns whose solution set is spanned by X and Y. (b) Find a system of three equations in four unknowns whose solution set is spanned by X and Y. (c) Find a system of four equations in four unknowns that has the set of vectors of the form Z + aX + bY as its general solution where Z = [1, 1, 1, 1]t . 2.1 Test the given matrices for linear dependence using the test for linear independence. Then find a basis for their span and express the other vectors (if there are any) as linear combinations of the basis elements. (f) [ 1 1 4 3 ] [ 2 1 3 0 ] [ 2 1 2 −1 ] [ 1 1 6 5 ] Requirements: need hand write step solution