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Unformatted text preview: Linear Algebra Problems Math 504 505 Jerry L. Kazdan Although problems are categorized by topics, this should not be taken very seriously since many problems fit equally well in several different topics. Notation: We occasionally write M ( n, F ) for the ring of all n n matrices over the field F , where F is either R or C . Basics 1. At noon the minute and hour hands of a clock coincide. a) What in the first time, T 1 , when they are perpendicular? b) What is the next time, T 2 , when they again coincide? 2. Which of the following sets are linear spaces? a) { X = ( x 1 ,x 2 ,x 3 ) in R 3 with the property x 1 2 x 3 = 0 } b) The set of solutions x of Ax = 0, where A is an m n matrix. c) The set of 2 2 matrices A with det( A ) = 0. d) The set of polynomials p ( x ) with integraltext 1 1 p ( x ) dx = 0. e) The set of solutions y = y ( t ) of y + 4 y + y = 0. 3. Which of the following sets of vectors are bases for R 2 ? a). { (0 , 1) , (1 , 1) } b). { (1 , 0) , (0 , 1) , (1 , 1) } c). { (1 , 0) , ( 1 , } d). { (1 , 1) , (1 , 1) } e). { ((1 , 1) , (2 , 2) } f). { (1 , 2) } 4. For which real numbers x do the vectors: ( x, 1 , 1 , 1), (1 ,x, 1 , 1), (1 , 1 ,x, 1), (1 , 1 , 1 ,x ) not form a basis of R 4 ? For each of the values of x that you find, what is the dimension of the subspace of R 4 that they span? 5. Let C ( R ) be the linear space of all continuous functions from R to R . a) Let S c be the set of differentiable functions u ( x ) that satisfy the differential equa tion u = 2 xu + c for all real x . For which value(s) of the real constant c is this set a linear subspace of C ( R )? 1 b) Let C 2 ( R ) be the linear space of all functions from R to R that have two continuous derivatives and let S f be the set of solutions u ( x ) C 2 ( R ) of the differential equation u + u = f ( x ) for all real x . For which polynomials f ( x ) is the set S f a linear subspace of C ( R )? c) Let A and B be linear spaces and L : A B be a linear map. For which vectors y B is the set S y := { x A Lx = y } a linear space? 6. Compute the dimension and find bases for the following linear spaces. a) Real antisymmetric 4 4 matrices. b) Quartic polynomials p with the property that p (2) = 0 and p (3) = 0. c) Cubic polynomials p ( x,y ) in two real variables with the properties: p (0 , 0) = 0, p (1 , 0) = 0 and p (0 , 1) = 0. d) The space of linear maps L : R 5 R 3 whose kernels contain (0 , 2 , 3 , , 1). 7. a) Compute the dimension of the intersection of the following two planes in R 3 x + 2 y z = 0 , 3 x 3 y + z = 0 . b) A map L : R 3 R 2 is defined by the matrix L := parenleftbigg 1 1 1 3 3 1 parenrightbigg . Find the nullspace (kernel) of L ....
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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.
 Spring '08
 Neely

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