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 Linear AlgebraLab 2


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1 Linear AlgebraLab 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. 3) Divide the polynomial p(x) = 2x 5 x 4 + 4x 3 + 3x 2 x + 1 by the polynomial q(x) = x 3 + x 2 x + 1 and find the remainder. 4) Using Horner s schema, find p(a) and p(b), where p(x) = 2x 4 3x 3 + 5x 2 x + 5 and a = 3, b = ) Given p(x) = x 5 4x 3 2x 2 + 3x + 2, find all roots of p(x) together with their multiplicity and express the given polynomial as a product as a product of real irreducible polynomials. Linear AlgebraLab 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) ) Determine if the matrix M = is a linear combination of A =, B =, C = ) 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 span{u 1, u 2, u 3 } = IR 3. 7) Given g = 5t 7t 2 in P(t), show that g span{g 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 AlgebraLab 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 ) In M 2 2 show that A =, B = and C = are Linearly Independent ) In M 2 2 show that A =, B = and C = are Linearly Dependent ) 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 ) 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.
2 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 = span{v 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 AlgebraLab 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) ) Given A = and B = , evaluate: A T, B A, B T, A 2, A B T, rank A, rank B. 4) Find λ IR such that the matrix A = λ λ ( 4 1 5) Find all matrices B such that B A = A B where A = ) Solve AX = B, where A =, B = has rank equal to two. ). Linear AlgebraLab 5 1) Using Gauss elimination method, find the inverse of the following matrices: A =, B = , C = , P = ) Solve the matrix equation AX = B where A = and B = ) 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 α 5) Determine the value of rank A for any possible α IR, where A = 0 7 4α. α ) Find the determinant of A by first reducing it to triangular form, A = Linear AlgebraLab 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 A =, B = , C = , P =
3 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 ) 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 AlgebraLab 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 = and B = x + 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 a a. 1) Determine if the following are linear transformations Linear AlgebraLab 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.
4 6) Given the linear transformation l : IR 4 IR 3 that has as associated matrix with respect to the standard bases A(l) = , write down the general form of l(x, y, z, w), find Ker(l), Im(l), their bases 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 AlgebraLab 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 AlgebraLab 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 ) Given the matrix M =, we define the linear transformation l : M 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. 4) Determine if A = 3 1, B = and C = are diagonalizable ) Determine if A = and B = are diagonalizable Notice that the given matrices have the same characteristic polynomial but they are not similar.
5 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 AlgebraLab 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 = B = C = ) 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 ) Determine for what values of k dim Im l k = ) Determine for what values of k dim Ker l k = ) For k = 0 is the transformation l 0 diagonalizable? Linear AlgebraLab 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), ( ) u, v M 2 3, u = A = ), v = B = 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.
6 Linear AlgebraLab 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 GramSchmidt algorithm to find an orthogonal basis of U, then find an orthonormal basis of U. 4) Given the symmetric matrix A = find an orthonormal real matrix P such that P t AP is diagonal (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 AlgebraLab 14 1) Find the parametric and canonical equation of the line p passing through the points A = [1, 0, 2] and B = [3, 1, 2]; check whether the point M = [7, 3, 1] lies on p. 2) Find the equation of the planes ρ and σ, verify that they are not parallel and find the parametric equation of the line p, intersection of ρ and σ, where ρ is the plane containing the point M = [1, 2, 3] and orthogonal to the vector n = (4, 5, 6); σ is the plane passing through the points A = [2, 5, 1], B = [2, 3, 3], and C = [4, 5, 0]. 3) Find the angle between the planes ρ and σ, if ρ passes through the points M 1 = [ 2, 2, 2], M 2 = [0, 5, 3] and M 3 = [ 2, 3, 4], and σ has equation 3x 4y + z + 5 = 0. 4) Find the distance between the point A = [8, 7, 1] and the plane with equation 2x + 3y 4z + 5 = 0. 5) Given A = [2, 9, 8], B = [6, 4, 2] and C = [7, 15, 7], show that AB and AC are perpendicular, then find D so that ABCD forms a rectangle. 6) Given the line p passing through the points A = [1, 2, 1] and B = [2, 1, 3], find the point P on p closest to the origin and the shortest distance from the origin to p. 7) Show that the planes x + y 2z = 1 and x + 3y z = 4 intersect in a line and find the distance between the point C = [1, 0, 1] and this line. 8) Find an equation for the plane through P = [1, 0, 1] and passing through the line of intersection of the planes x + y 2z = 1 and x + 3y z = 4. 9) Find an equation for the plane passing through P = [6, 0, 2] and perpendicular to the line of intersection of the planes x + y 2z = 4 and 3x 2y + z = 1. 10) Find an equation for the plane passing through the point A = [1, 0, 2] and containing the line p with vector equation X = [1, 1, 1] + t(3, 2, 0), t IR. 11) Given the line p through A = [1, 2, 1] and B = [3, 1, 2] and the line q through C = [1, 0, 2] and D = [2, 1, 3], prove that the distance between p and q is ) Find the point R symmetric to P = [ 4, 5, 8] with respect to the line p through A = [9, 4, 10] and B = [ 6, 1, 1]. 13) A line with directional vector v = (0, 9, 1) intersects lines p and q, find the coordinates of the points of intersection, where x 8 p: 5 = y 5 1 = z 1, and q: x 1 = y 1 2 = z ) Find the distance of the point Q = [3, 2, 1] from the plane containing the lines p and q, where x+1 p: 1 = y 3 2 = z 2 2, and q: X = [0, 4, 2] + t(1, 1, 0), t IR.
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