# 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 Algebra-Lab 2

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1 Linear Algebra-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. 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 Algebra-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) ) 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 Algebra-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 ) 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 Algebra-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) ) 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 Algebra-Lab 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 Algebra-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 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 Algebra-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 = 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 Algebra-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.

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 Algebra-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-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 ) 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 Algebra-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 = 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 Algebra-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), ( ) 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 Algebra-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 = 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 Algebra-Lab 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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