la - Linear Algebra Problems Math 504 505 Jerry L Kazdan...

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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 , 0 } 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
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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 anti-symmetric 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 , 0 , 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 . 8. If A is a 5 × 5 matrix with det A = 1, compute det( 2 A ). 9. Does an 8-dimensional vector space contain linear subspaces V 1 , V 2 , V 3 with no com- mon non-zero element, such that a). dim( V i ) = 5, i = 1 , 2 , 3? b). dim( V i ) = 6, i = 1 , 2 , 3? 10. Let U and V both be two-dimensional subspaces of R 5 , and let W = U V . Find all possible values for the dimension of W .
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