Third Homework Assignment (Due in class, Friday, February 3) Practice Exercises Section 2.5: 6, 8, 10, 12, 14, 15-18 Section 3.1: 2, 10, 14, 16, 22, 23-27 Section 3.2: 2, 5, 12, 17-23 Practice Problems Section 2.5: 3 Section 3.1: 1, 4, 5 Section 3.2: 1, 3, 5 Graded Homework Exercises, Four Points Each x1 x2 2.5.1. Let V = { x3 : x1 x2 x3 + x4 = 0} and B = (v1 , v2 , v3 , v4 ) where x4 2 2 1 1 3 5 2 1 v1 = 1 , v2 = 1 , v3 = 2 , v4 = 3 . 5 2 1 4 Find a subsequence of B which is a basis for V . If there is no basis say NONE. 2 1 1 1 2 3 2.5.2. Let W = Span( 1 , 1 , 3) and C = (v1 , v2 , v3 , v4 , v5 ) where 8 4 3 2 1 2 0 1 1 1 1 1 0 v1 = 1 , v2 = 1 , v3 = 1 , v4 = 2 , v5 = 4 . 7 4 2 2 2 Find a subsequence of C which is basis of W . If there is no subsequence write NONE. 1 1 2 1 2 3 2.5.3. Let U = Span(D) where D = ( 1 , 1 , 1). The sequence D is 1 2 4 2 3 linear independent and therefore a basis of U . The vector v = 1 U. 2 Find the coordinate vector of v with respect to D. 1 3.1.1. T : R3 R4 is a linear transformation and 1 2 1 1 0 0 1 3 2 T 0 = , T 1 = , T 0 = . 2 2 3 0 0 1 2 1 0 3 Determine T 2 . 1 1 3.1.2. Let A = 1 2 1 2 3 2 1 2 2 3 1 1. Compute A 1 . 3 1 3.1.3. T : R3 R3 is a linear transformation and 1 1 2 2 1 1 T 1 = 2 , T 3 = 5 , T 1 = 0 . 1 3 2 7 2 2 Determine the standard matrix of T . 2 1 2 2 3 3.2.1. Let A = 3 2 1 1 independent sequence of 1 0 1 0 1 1 . Express null(A) as a span of a linearly 1 3 1 5 2 2 vectors. 1 1 1 0 2 1 2 0 1 3 3.2.2. Let B = 2 3 2 1 5 . Express the solution set of the ma3 2 3 1 5 1 1 trix equation Bx = 1 in the form p + S where S is a subspace, S = 4 Span(v1 , . . . , vk ) with (v1 , . . . , vk ) a linearly independent sequence of vectors, and p is an appropriate 5-vector. 3.2.3. Let 1 1 C= 1 2 1 2 2 3 1 0 1 2 2 0 1 3 ,D = 2 2 1 2 1 3 3 2 3 3 6 3 1 5 . 5 1 Determine the set of all x R4 such that Cx = Dx. Express in the from Span(v1 , . . . , vk ) with (v1 , . . . , vk ) a linearly independent sequence of vectors. 3