Mid-term Exam #1 MATH 205, Fall 2014

Transcription

1 Mid-term Exam # MATH, Fall Name: Instructions: Please answer as many of the following questions as possible. Show all of your work and give complete explanations when requested. Write your final answer clearly. No calculators or cell phones are allowed. This exam has problems and points. Good luck! Problem Possible Points Points Earned TOTAL

2 . ( points) Consider the following linear system of equations: 8 >< >: x + x + x 8 x x x x + x x + 8x x + x + x (a) ( points) Find the reduced echelon form of the augmented matrix associated with this system. (b) ( points) Write the general solution to the system in parametric vector form. (c) ( points) Write the general solution to the corresponding homogeneous system of equations in parametric vector form.

3 (a) 8 8 r! r + r r! r + r r! r + r r $ r r! (/)r r! r + r r! r + r (b) x (c) x x x x x x x x x t 8 + t, t in R, t in R

4 . ( points) Let u, v. (a) ( points) Is the set {u, v} linearly independent or dependent? Justify your answer. (b) ( points) Find all values of h such that w is in Span{u, v}. h (a) The set u, v is linearly independent because by inspection I see that u is not a multiple of v. (b) Find h so that there exist c,c such that c u + c v w. Write this vector equation as an augmented matrix and row reduce to find h so that the system is consistent: h r! r + r r $ r r! r + r h h h h From the last matrix above we see that the vector equation will have asolutionwhenh /. Then w is in Span {u, v} when h /..

5 . ( points) Let T (x) :R! R be given by T " x x #! " x +x x +x #. (a) ( points) Show, using the definition of linear transformation, that T is a linear transformation. (b) ( points) Find A such that T (x) Ax. ShowthatA A. (c) ( points) Is T amappingfromr onto R? Give a one sentence explanation supporting your answer. If T is not onto, find a vector b that is not in the range of T. (a) Let u of T. apple apple u v, v u v and c be a real number. Check two properties i. Show that for all u, v in R, T (u + v) T (u)+t(v): apple u + v T (u + v) T u + v apple u +v +u +v u +v +u +v apple u +u + v +v + v +v u +u apple u +u u +u + T (u)+t (v) apple v +v v +v ii. Show that for all u in R and all c in R, T (cu) ct (u): apple cu T (cu) T cu c apple cu +cu cu +cu " c(u +u ) c(u +u ) apple u +u u +u ct (u) #

6 apple / / (b) A / / A apple apple / / / / / / / / apple /+/ /+/ /+/ /+/ apple / / A / / (c) Tapple is not an onto mapping. The reduced echelon form of the matrix A is. This matrix does not have a pivot in every row so therefore T is not a mapping from R onto R. Alternative answer: The columns of A do not span R,thereforeT is not a mapping from R onto R. apple The vector b is not in the range of T (there are many correct answers). If I augment the matrix A with the vectorapple b and row reduce, the reduced echelon form of the augmented matrix is which shows that the system Ax b is inconsistent. This, in turn means that there is no vector x in R such that T (x) b.

7 . ( points) (a) ( points) Consider the following system of equations: x + hx x +x k. Choose h and k so that the system has i. no solution; ii. auniquesolution; iii. many solutions. (b) ( points) Let A be an n n matrix. Explain why the columns of A are linearly dependent when det A. Write your answer in complete sentences. (a) Build the augment matrix associated the the system and perform the row reduction algorithm to obtain echelon form: apple apple h h r k! r + r () h k i. From (), the system will have no solution if h andk. Or, if h / andk. Thiswillmakethesystemrepresentedby () inconsistent. ii. From (), the system will have a unique solution if h and k isanyrealnumber.or,ifh / andk is any real number. iii. From (), the system will have many solutions is if there is a free variable, meaning that h andk. Or, equivalently, if h / andk. (b) Suppose that det A. Then A is not invertible. By the Invertible Matrix Theorem, this means that the homogenous equation Ax has non-trivial solutions. This, in turn, means that the columns of A are linearly dependent.

8 . ( points) (a) ( points) Compute det (B ), where B (b) ( points) Show that if A is invertible, then det A det A. (a) By the multiplicative property of determinants, we know that det (B ) det B.SowecalculatedetBusing the method of cofactors, expanding across the first row: det B det apple + +. Then det B ( ). apple det (b) By the multiplicative property of determinants, det AA )deta det A.. + apple det IalsoknowthatAA I n, the n n identity matrix, and det I n (this can be seen because I n is a diagonal matrix). So det A det A. Solving for det A, wegetdeta, where is it okay to divide by det A det A because we know that det A bythefactthata is invertible. 8