s11_mthsc853_hw05

# s11_mthsc853_hw05 - b is in the column space of A(b...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 5 Due Monday, February 21st, 2011 (1) Let T : X U , with dim X = n and dim U = m . Show that there exist bases B for X and B 0 for U such that the matrix of T in block form is M = ± I k 0 0 0 ² , where I k is the k × k identity matrix. (2) Consider the linear map T : R 3 R 3 with matrix representation 1 - 1 0 0 2 - 2 - 3 0 3 with respect to the standard basis. What is the matrix representation of T with respect to the basis 1 - 1 0 , 0 1 - 1 , 1 0 1 ? (3) Let P n = { p ( x ) R [ x ] | deg( p ( x )) < n } . (a) Show that the map T : P 3 → P 4 given by T ( p ( x )) = 6 Z x 1 p ( t ) dt is linear. Indicate whether it is 1–1 or onto. (b) Let B 3 = { 1 ,x,x 2 } be a basis for P 3 and let B 4 = { 1 ,x,x 2 ,x 3 } be a basis for P 4 . Find the matrix representation of T with respect to these bases. (4) Let T be a matrix over a ﬁeld K . (a) Prove that if T has a left inverse, then Tx = u has at most one solution. (b) Prove that if T has a right inverse, then Tx = u has at least one solution. (c) What are the possibilities for the rank of T if it has a left inverse? What if it has a right inverse? (5) Let A be an m × n matrix, and b R n . (a) Prove that Ax = b is solvable if and only if
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Unformatted text preview: b is in the column space of A . (b) Identify geometrically, as clearly as you can, the subset of 3-dimensional Euclidean space of R 3 that corresponds to the column space of the matrix A = 2-2 3 5-2 4 4 . (6) Find necessary and suﬃcient conditions on the entries u 1 ,u 2 ,u 3 ,u 4 under which the fol-lowing system of linear equations will have at least one solution over Z 5 = { , 1 , 2 , 3 , 4 } , and give the number of solutions in case the conditions are met. 1 2 1 1 2 1 3 1-1-1 2 3 2-1 x 1 x 2 x 3 x 4 = u 1 u 2 u 3 u 4 (7) Consider the system of linear equations Tx = u , where T = 2-2 3 5-2 4 4 , u = 1-1 1 . (a) Solve Tx = u over Q . (b) Solve Tx = u over Z 2 ....
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