Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4) 2x + 3y + 6z = 10 3x + 6y + 10z = 17 x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 x + 2y 3z = 1 1.3) 2x + 5y 8z = 4 3x + 8y 13z = 7 2) Write down all solutions of the following system as p IR obtains all possible real values 2x y z = p x y = p x + 2y + z = 1. Linear Algebra and its Applications-Lab 2 1) Determine if the vector v = (2, 5, 3) in IR 3 is a linear combination of u 1 = (1, 3, 2), u 2 = (2, 4, 1) and u 3 = (1, 5, 7). ( ) ( ) ( ) ( ) 4 7 1 1 1 2 1 1 2) Determine if the matrix M = is a linear combination of A =, B =, C =. 7 9 1 1 3 4 4 5 3) Is W = (a, b, c) : a 0} a linear subspace of IR 3? 4) Given the linear space P(t) of all polynomials, determine if the following are linear subspaces - all polynomials with integer coefficients - all polynomials with even powers of t - all polynomials with degree greater or equal to six. 6) Given u 1 = (1, 1, 1), u 2 = (1, 2, 3) and u 3 = (1, 5, 8), show that spanu 1, u 2, u 3 } = IR 3. 7) Given g = 5t 7t 2 in P(t), show that g spang 1, g 2, g 3 } where g 1 = 1 + t 2t 2, g 2 = 7 8t + 7t 2, g 3 = 3 2t + t 2. Linear Algebra and its Applications-Lab 3 1) Show that v 1 = (2, 1, 3), v 2 = (3, 2, 5), v 3 = (1, 1, 1) form a basis of IR 3. Find coordinates of v = (7, 6, 14) with respect to this basis. ( ) ( ) ( ) 1 1 1 0 1 1 2) In M 2 2 show that A =, B = and C = are Linearly Independent. 1 1 0 1 0 0 ( ) ( ) ( ) 1 2 3 1 1 5 3) In M 2 2 show that A =, B = and C = are Linearly Dependent. 3 1 2 2 4 0 4) What is the standard basis and the dimension of M n m for every n, m IR? 5) Consider the subset of M 2 2 formed by the symmetric matrices (a ij = a ji, for every i, j). Show that this is a subspace of M 2 2 and find a basis. 6) Show that 1, t, t 2 } and 1, 1 t, (1 t) 2 } are both basis of P 2 (t). Show that dim P n (t) = n + 1. 7) Determine for what values of α IR the vectors v 1 = (1, α, 1), v 2 = (0, 1, α), v 3 = (α, 1, 0) are linearly dependent. 8) In IR 4 are given the vectors v 1 = (0, 3, 2, 4), v 2 = (0, 1, 1, 3), v 3 = (0, 0, 1, 5), v 4 = (0, 5, 4, 10). Find a basis of the space M = spanv 1, v 2, v 3, v 4 }. Show that B = (0, 2, 1, 1), (0, 1, 0, 2)} is also a basis of M. 9) Consider M = p(t) P 3 (t) : p(t) is divisible by (t 1)}. Prove that M a linear subspace of P 3, find a basis and dimension of M.
Linear Algebra and its Applications-Lab 4 1) Show that if v 1, v 2, v 3, v 4 } is a basis of a linear space L then v 1, v 1 v 2, v 1 v 2 v 3, v 1 v 2 v 3 v 4 } is also a basis of L. (Use coordinates) 2) Given p 1 = 1 x 2, p 2 = 1 + x, p 3 = 1 + x + x 3, extend p 1, p 2, p 3 } to a basis of P 3. (Use coordinates) 1 0 2 1 0 2 1 1 3) Given A = and B = 2 0 0 1 1 1 1 1, evaluate: A 1 1 1 0 T, B A, B T, A 2, A B T, rank A, 2 1 2 0 1 2 1 1 rank B. 4) Find λ IR such that the matrix A = 1 2 3 1 λ 1 1 1 λ ( 4 1 5) Find all matrices B such that B A = A B where A = 3 2 ( ) ( ) 2 1 1 1 0 6) Solve AX = B, where A =, B =. 2 1 1 0 1 has rank equal to two. ). Linear Algebra and its Applications-Lab 5 1) Using Gauss elimination method, find the inverse of the following matrices: 1 1 2 1 0 1 2 1 A =, B = 1 2 3 1 1 2, C = 1 0 1 1 1 1, P = 1 1 1 1 4 2. 0 0 1 2 2 1 1 2 1 1 1 1 1 0 0 1 0 2) Solve the matrix equation AX = B where A = 1 1 0 1 1 1 and B = 1 2 3 2 3 4. 1 0 1 3 4 5 3) Find the determinant of the matrices given in 1). 4) Given the matrix A = 1 a a2 1 b b 2, verify that det A = (c b)(b a)(c a). 1 c c 2 16 0 4α 5) Determine the value of rank A for any possible α IR, where A = 0 7 4α. α 0 1 1 0 2 3 0 1 3 4 6) Find the determinant of A by first reducing it to triangular form, A =. 3 2 0 5 4 3 5 0
Linear Algebra and its Applications-Lab 6 1) For each of the following matrices find the determinant, the classical adjoint matrix and the inverse using the formula A 1 = 1 det A adj A 1 1 2 1 0 1 2 1 A =, B = 1 2 3 1 1 2, C = 1 0 1 1 1 1, P = 1 1 1 1 4 2. 0 0 1 2 2 1 1 2 1 1 1 1 1 0 0 1 0 2) Use Cramer s rule to solve the system x + 2y + z = 3 2x + 5y z = 4 3x 2y z = 5 [x = 2, y = 1, z = 3] 3) Determine for what value of the parameter a the following system has a unique solution, infinitely many solutions or no solution. Write down all solutions if any. 3.1) x + ay 3z = 5 ax 3y + z = 10 x + 9y 10z = a + 3 3.2) x + y + az = a x + ay + z = 1 ax + y + z = 1 x + y az = 1 3.3) x 2y + 3z = 2 x + ay z = 1 Linear Algebra and its Applications-Lab 7 1) Write down all solutions of the following system as the sum of a particular solution of the nonhomogeneous system with the linear space of solutions of the homogeneous system 1.1) x + y + z = 2 3x y + z = 0 1.2) x 2y + z w = 1 x + y z + w = 2 2x y + z w = 1 1.3) 2) Write down all solutions of the following system for any possible value of α, β IR 2x + y + z = 7 αx + 2y z = 2 3x + y + 2z = β 3) Solve the matrix equation XA = (X + I)B where I is the identity matrix, A = 6 9 1 4 2 4 and B = 5 7 1 2 0 3. 1 0 1 1 1 2 4x + y + z + 4w = 33 5x 4y + 2z w = 18 2x 3y + z 2w = 1 x + 2y + 3w = 16 4) Determine for what value of a IR the following matrix is regular and for those value find the inverse matrix A = a + 1 0 0 0 a 1. 0 1 a
1) Determine if the following are linear transformations Linear Algebra and its Applications-Lab 8 1.1) f : IR 2 IR, f(x, y) = x + y, 1.2) f : IR 2 IR, f(x, y) = x + 1, 1.3) f : IR 2 IR, f(x, y) = xy. 1.4) f : P 2 IR 2, f(ax 2 + bx + c) = (a + b, b + c), 1.5) f : IR 2 P 1, f(a, b) = b + a 2 x. 2) Given the linear transformation l : IR 3 P 1, l(a, b, c) = b c + (2a c)x, find the matrix associated with l with respect to the standard bases, evaluate l(2, 1, 1), find ker l its basis and dimension. Is l surjective? 3) Given the linear transformation l : IR 2 IR 2, l(x, y) = (2x 2y, x + y), write the matrix associated to l with respect to the standard basis of IR 2, find Ker l, Im l, its bases and dimensions. Find all vectors of IR 2 that are mapped to (4, 2). 4) Given l : IR 3 IR 3, l(x 1, x 2, x 3 ) = (x 1 + 2x 2 + 3x 3, 4x 1 + 5x 2 + 6x 3, x 1 + x 2 + x 3 ), find Ker(l), Im(l), their bases and dimensions. 5) Given l : IR 3 IR 2, l(x 1, x 2, x 3 ) = (2x 1 x 2 +3x 3, x 1 +x 2 +x 3 ), find Ker(l), Im(l), their bases and dimensions. 6) Given the linear transformation l : IR 4 IR 3 that has as associated matrix with respect to the standard bases A(l) = 1 1 1 1 1 0 2 1, write down the general form of l(x, y, z, w), find Ker(l), Im(l), their bases 1 1 3 3 and dimensions. ( ) a b 7) Given l : M 2 2 P 3 defined by l(a) = l = a + (2a b)x + (b + c)x c d + (a b + c + d)x 3, find the matrix associated with l with respect to the standard bases. Is l an isomorphism? Linear Algebra and its Applications-Lab 9 1) In IR 3 are given the standard basis C = (1, 0, 0), (0, 1, 0), (0, 0, 1)} and the bases B = (1, 1, 0), (0, 1, 0), (0, 1, 1)}, D = (1, 0, 1), (1, 1, 1), (0, 1, 1)}. Find the transition matrices P B C, P D C, P B D, P D B. 2) Given l : IR 2 IR 2 such that l(x, y) = (4x 2y, 2x + y), find the matrix associated to l with respect to the basis F = (1, 1), ( 1, 0)}. 3) Given the linear transformation l : IR 3 IR 3 defined by l(1, 2, 3) = ( 3, 8, 3), l(1, 1, 0) = (1, 5, 2) and such that Ker l = span(1, 1, 1)}, find the matrix associated with l with respect to the standard bases, find Im l, its basis and dimension. Find all v IR 3 such that l(v) = (2, 3, 1). 4) Given the linear transformation l λ : IR 3 IR 3, l λ (x, y, z) = (2x+y, y z, 2y+λz), write the matrix associated to l λ with respect to the standard basis of IR 3, find Ker l, Im l, its bases and dimensions for every possible value of λ IR. Does there exist a value of λ for which l λ is an isomorphism?. 5) Given l : IR 4 IR 3 such that l(1, 1, 1, 0) = (0, 0, 0), l(1, 2, 1, 2) = ( 1, 3, 1), l(1, 0, 0, 1) = (0, 0, 0), l(1, 1, 1, 1) = (5, 8, 2), find the matrix A associated with l with respect to the standard bases. Find all v IR 4 such that Av = (13, 21, 5). 6) Given the transformation l h : IR 3 IR 3 defined by l h (x, y, z) = (x hz, x + y hz, hx + z), where h IR is a parameter. Find, for all possible values of h, Ker(l h ), Im(l h ), their bases and dimensions. Determine l 1 h (1, 0, 1) = (x, y, z) IR3 : l h (x, y, z) = (1, 0, 1)}.
Linear Algebra and its Applications-Lab 10 1) Given the linear transformations f : IR 4 IR 3, f(x, y, z, t) = (x t, x + y, z + y) and g : IR 3 IR 4, g(x, y, z) = (z x, y, y, x + t), 1.1) find Ker f, Im f,ker g, Im g, 1.2) solve g(x, y, z) = (h, 1, h, 1) for any possible h IR, 1.3) write down the general form of f g, g f and determine if they are isomorphisms, 1.4) are f g, g f diagonalizable? find all respective eigenvalues and a basis for each corresponding eigenspace. ( ) 1 2 2) Given the matrix M =, we define the linear transformation l : M 3 4 2 2 M 2 2 by l(x) = M X. Is l an isomorphism? If possible, find its inverse. 3) Given the transformation l : IR 3 IR 3, l(x, y, z) = (y + z, x z, x + y + z), determine if it is invertible and, if possible, find its inverse. ( ) ( ) 3 1 1 1 4) Determine if A =, B = and C = 2 1 1 0 3 1 are diagonalizable. 1 1 2 1 0 0 2 5) Determine if A = 1 3 3 3 1 1 3 5 3 and B = 7 5 1 are diagonalizable. 6 6 4 6 6 2 Notice that the given matrices have the same characteristic polynomial but they are not similar. 6) Given the transformation l : IR 3 IR 3, l(x, y, z) = (2x + y, y z, 2y + 4z), determine if it is invertible and, if possible, find its inverse. Find all eigenvalues and a basis of each eigenspace. Is l diagonalizable? Linear Algebra and its Applications-Lab 11 1) Given the linear transformation l : IR 2 IR 2, l(x, y) = (x y, x+3y), write the matrix A (l,b,b) associated to l with respect to the basis B = (1, 2), (2, 2)}. Determine if there exists a basis S of IR 2 such that A (l,s,s) is diagonal. 2) Given the linear transformation l : IR 3 IR 3, l(x, y, z) = (2x+y+3z, x+3y+z, x) find a basis B of eigenvectors of l, such that the matrix D associated to l with respect to B is diagonal. Verify that P 1 AP = D, where P is the transition matrix from basis B to the canonical one, and A is the matrix associated to l with respect to the canonical basis. 3) Diagonalize the following matrices (for each eigenvalue find a basis of the corresponding eigenspace) A = 2 1 0 1 2 0 B = 2 1 1 1 2 1 C = 0 1 4 1 1 1. 1 1 1 2 2 2 1 0 1 4) Consider the following system kx + y z = k (1 k)y + z = h + k with h, k real parameters. y + (1 k)z = 2h + 1 Determine for what values of h, k IR the system has one unique solution, no solutions or infinitely many solutions. 5) Write the matrix A k associated to the system in 3), so that the system can be written in the form A k x y = k h + k z 2h + 1 Consider the linear transformation l k IR 3 IR 3 associated to A k with respect to the canonical basis of IR 3. 5.1) Determine for what values of k dim Im l k = 3. 5.2) Determine for what values of k dim Ker l k = 2. 5.3) For k = 0 is the transformation l 0 diagonalizable?
Linear Algebra and its Applications-Lab 12 1) Given the vectors v = (1, 5), u = (3, 4) in IR 2, with the standard inner product, find < u, v >, u, v. 2) In C[0, 1] with the standard inner product, i.e. < f, g >= g(t) = 3t 2, h(t) = t 2 2t 3. Find < f, g >, < f, h >, f, g. 3) Find cos θ = <u,v> u v, where θ is the angle between the vectors v, u, and 3.1) u, v IR 4, u = (1, 3, 5, 4), v = (2, 3, 4, 1), ( 9 8 7 3.2) u, v M 2 3, u = A = 6 5 4 ), v = B = 1 0 ( ) 1 2 3. 4 5 6 f(t)g(t) dt, consider the functions f(t) = t + 2, 4) Verify that cos t, sin t are orthogonal vectors in C[0, 2π], with respect to the standard inner product. 5) Find k IR such that u = (1, 2, k, 3) and v = (3, k, 7, 5) are orthogonal in IR 4. 6) Given, in IR 5, W = span(1, 2, 3, 1, 2), (2, 4, 7, 2, 1)}, find a basis of the orthogonal complement W. 7) In IR 4 is given the space W = span(1, 2, 3, 1)}, find an orthogonal basis of the orthogonal complement W. Linear Algebra and its Applications-Lab 13 1) In IR 4 is given the set S = (1, 1, 0, 1), (1, 2, 1, 3), (1, 1, 9, 2), (16, 13, 1, 3)}. Show that S is orthogonal, and it forms a basis of IR 4. Find the coordinates of v = (a, b, c, d) with respect to S. 2) Find the Fourier coefficient c = <v,w> w and the projection of v = (1, 2, 3, 4) along w = (1, 2, 1, 2) with 2 respect to the standard inner product in IR 4. 3) In IR 4 consider U = span(1, 1, 1, 1), (1, 1, 2, 4), (1, 2, 4, 3)}. Use the Gram-Schmidt algorithm to find an orthogonal basis of U, then find an orthonormal basis of U. 4) Given the symmetric matrix A = 2 1 1 1 2 1 find an orthonormal real matrix P such that P t AP is diagonal. 1 1 2 (Remember that P is constructed with orthonormal eigenvectors of A.) 5) Prove that every symmetric 2 2 matrix is diagonalizable. 6) Verify that the linear transformation l : IR 3 IR 3, defined by l(x, y, z) = (x + 3y + 4z, 3x + y, 4x + z) is symmetric, prove that it is diagonalizable and there exists a basis of IR 3 made of three orthogonal eigenvectors of l.
Linear Algebra and its Applications-Lab 14 1) For the following separable differential equations find the solution of the given Cauchy problem and its domain. 1.1) y = y2, y(1) = 2. x 1.2) y = 6y 2 x, y(1) = 1 9. 1.3) y = 2 y e x, y(0) = 4. 1.4) y = (y 1) cos x, y(π) = 0. 1.5) x = 2t 1 x 2, x( π) = 1 2. 1.6) y = 2 y, a) y(0) = 1, b) y(0) = 0. 1.7) x = 1 x2 2tx, a) x(1) = 1 2, b) x(2) = 2, c) x( 1) = 1, d) x( 2) = 2, e) x( 8) = 1 2. 2) Solve the Cauchy problem y = (t + y) 2, y(0) = 0 by first performing a substitution to transform the equation in a separable one. [Hint:z(t) = t + y(t)] Linear Algebra and its Applications-Lab 15 1) Find the solution of the given Cauchy problem and its domain. 1.1) ty + 2y = t 2 t + 1, y(1) = 1 2. 1.2) y + 1 x y = 1, a) y(1) = 1, b) y(0) = 3, c) y( 1) = 2,. x2 1.3) x = 2t(t 2 + x), x(0) = 1. 1.4) x = x cot(t) + 2t sin 2 t, x( 7 2π) = 1. 1.5) y + y x 1 = 3x + 3, y(0) = 3. 1.6) 2y = 12x 2 + 3x 2 y 2, y(0) = 0. 1.7) x + x tan t = 3 tan t, x(0) = 1. 1.8) y y e x + 1 = ex, y(0) = 2. 1.9) y = y x + 4x2, y( 1) = 0.
Linear Algebra and its Applications-Lab 16 1) Given the equation y 2 x y + 2 x 2 y = 2x, verify that x, x 2 } is a fundamental system of solutions. Use variation of constants to find the solution satisfying the initial conditions y(1) = 3, y (1) = 6. 2) Given the equation y 3 x y + 6 x 2 y 6 x 3 y = 6x, verify that x, x 2, x 3 } is a fundamental system of solutions. Use variation of constants to find the solution satisfying the initial conditions y(1) = 2, y (1) = 6, y (1) = 14. 3) Find a fundamental system of solutions for the given linear homogeneous differential equation with constant coefficient. 3.1) y + y 6y = 0 3.2) y + 4y + 4y = 0 3.3) y 3y = 0 3.4) y IV + 4y = 0 3.5) y 4y + 13y = 0 3.6) y 4y + 5y 2y = 0 4) Find the solution of y y + y y = 0, with initial conditions y(0) = 1, y (0) = 0, y (0) = 0. 5) Solve the following differential equation. 5.1) x 2x + x = et t. 5.2) x + x = 1 sin t, x( π 2 ) = 0, x ( π 2 ) = 0. 5.3) y + y = cot x. 5.4) y 2y + y = ex 1+x 2, y(0) = 1, y (0) = 2. Linear Algebra and its Applications-Lab 17 1) Given the linear ODE with constant coefficients y 4y = g(x), find a fundamental system of solutions and guess the form of a particular solution in each of the following cases: 1.1) g(x) = xe 2x 1.2) g(x) = 5 sin(2x) 1.3) g(x) = 7x 2 + 3 1.4) g(x) = xe 2x cos(x) 2) Find a general solution for x + x = 4 sin t. 3) Find a general solution for y 2y + y = e x 2e 2x. 4) Solve the Cauchy problem y + y = 4x + 2xe x, y(0) = 1 2, y (0) = 1 2. 5) Solve the Cauchy problem y + 3y 4y = e x + 1, y(0) = 0, y (0) = 2. 6) Solve the Cauchy problem y 2y = 4e 2x 5e x + 2, y(0) = 0, y (0) = 2. 7) Solve the Cauchy problem y + 3y = sin x + 1, y(0) = 0, y (0) = 0. 8) Using the elimination method solve the system y 1 = 2y 1 y 2, y 1 (0) = 1 y 2 = 4y 1 3y 2, y 2 (0) = 2 Linear Algebra and its Applications-Lab 18 1) Solve the following systems (without using the elimination method!) x 1.1) = 3x 2y x y 1.2) = 2x + y = 2x 2y y 1.3) = x + 2y x = 3x + y y = x + y 2) Solve the Cauchy problem x = 3x 2y, x(0) = 0 y = 2x 2y, y(0) = 0. Are the solutions bounded on [0, + ]? 3) Solve the nonhomogeneous systems x 3.1) = x + 4y + e 3t y = x + 3y 3.2) x = x + 4y + 2 + e 3t y = x + 3y 1 3.3) x = x + 4y + cos t y = x + 3y 4) Solve the Cauchy problem x = x + 2y + e t, x(0) = 0 y = 2x + y, y(0) = 0.