Download Linear Algebra-Lab 1 1) Use Gaussian elimination to solve the

Linear Algebra-Lab 1
1) Use Gaussian elimination to solve the following systems

 x1 + x2 − 2x3 + 4x4 = 5
1.1) 2x1 + 2x2 − 3x3 + x4 = 3

3x1 + 3x2 − 4x3 − 2x4 = 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 + 2y − 3z = 1
1.3) 2x + 5y − 8z = 4

3x + 8y − 13z = 7

 x + 3y − 2z + 5t = 4
1.5)
2x + 8y − z + 9t = 9

3x + 5y − 12z + 17t = 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) = 2x5 − x4 + 4x3 + 3x2 − x + 1 by the polynomial q(x) = x3 + x2 − x + 1 and find
the remainder.
4) Using Horner’s schema, find p(a) and p(b), where p(x) = 2x4 − 3x3 + 5x2 − x + 5 and a = 3, b = 21 .
5) Given p(x) = x5 − 4x3 − 2x2 + 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 IR3 is a linear combination of u1 = (1, −3, 2), u2 = (2, −4, −1) and
u3 = (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 IR3 ?
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 u1 = (1, 1, 1), u2 = (1, 2, 3) and u3 = (1, 5, 8), show that span{u1 , u2 , u3 } = IR3 .
7) Given g = 5t − 7t2 in P(t), show that g ∈ span{g1 , g2 , g3 } where g1 = 1 + t − 2t2 , g2 = 7 − 8t + 7t2 ,
g3 = 3 − 2t + t2 .
Linear Algebra-Lab 3
1) Show that v1 = (2, 1, −3), v2 = (3, 2, −5), v3 = (1, −1, 1) form a basis of IR3 .
Find coordinates of v = (7, 6, −14) with respect to this basis.
1 1
1 0
1 1
2) In M2×2 show that A =
,B=
and C =
are Linearly Independent.
1 1
0 1
0 0
1 2
3 −1
1 −5
3) In M2×2 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 Mn×m for every n, m ∈ IR?
5) Consider the subset of M2×2 formed by the symmetric matrices (aij = aji , for every i, j). Show that this is
a subspace of M2×2 and find a basis.
6) Show that {1, t, t2 } 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 v1 = (1, α, 1), v2 = (0, 1, α), v3 = (α, 1, 0) are linearly
dependent.
8) In IR4 are given the vectors v1 = (0, 3, −2, 4), v2 = (0, 1, −1, 3), v3 = (0, 0, 1, −5), v4 = (0, 5, −4, 10). Find a
basis of the space M = span{v1 , v2 , v3 , v4 }. 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 {v1 , v2 , v3 , v4 } is a basis of a linear space L then {v1 , v1 ]v2 , v1 ]v2 ]v3 , v1 ]v2 ]v3 ]v4 } is also a
basis of L. (Use coordinates)
2) Given p1 = 1 − x2 , p2

1 0
0 2
3) Given A = 
1 −1
1 2
rank B.
= 1 + x, p3 = −1 + x + x3 , extend {p1 , p2 , p3 } to a basis of P 3 . (Use coordinates)



2 −1
2 0 0 −1
1 1 
 and B =  1 1 1 1 , evaluate: AT , B · A, BT , A2 , A · BT , rank A,
1 0
2 −1 2 0
1 −1


3
−1  has rank equal to two.
−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
1
4) Find λ ∈ IR such that the matrix A =  1
1
−2
λ
1
Linear Algebra-Lab 5
1) Using Gauss elimination method, find the inverse of the following matrices:
1
0
A=
0
0

1
1
0
0
2
2
1
1

−1
−1 
,
2
0

1
B =  −1
2

2 3
1 2 ,
1 −1

1
C = 1
2

1 −1
2) Solve the matrix equation AX = B where A =  −1 1
1
0
0
−1
1

1
1 ,
1

1
P = 1
1


1
0
−1  and B =  2
3
1
2
3
4

1 −1
−4 2 .
−1 1

3
4 .
5
3) Find the determinant of the

1

4) Given the matrix A = 1
1
matrices given in 1).

a a2
b b2 , verify that det A = (c − b)(b − a)(c − a).
c c2


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-Lab 6
1) For each of the following matrices find the determinant, the classical adjoint matrix and the inverse using the
formula A−1 = det1 A adj A
1
0
A=
0
0

1
1
0
0
2
2
1
1

−1
−1 
,
2
0

1
B =  −1
2

2 3
1 2 ,
1 −1

1
C = 1
2
0
−1
1

1
1 ,
1

1
P = 1
1
1
−4
−1

−1
2 .
1
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.

 x + ay − 3z
3.1)
ax − 3y + z

x + 9y − 10z
=
5
=
10
= a+3

 x + y + az
3.2) x + ay + z

ax + y + z
= a
= 1
= 1

 x + y − az
3.3) x − 2y + 3z

x + ay − z
=
=
=
1
2
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
3x − y + z

 x − 2y + z − w
x+y−z+w
1.2)

2x − y + z − w
= 2
= 0
= 1
= 2
= 1

4x + y + z + 4w


5x − 4y + 2z − w
1.3)

 2x − 3y + z − 2w
x + 2y + 3w
= 33
= 18
= 1
= 16
2) Write down all solutions of the following system for any possible value of α, β ∈ IR

 2x + y + z
αx + 2y − z

−3x + y + 2z
= 7
= 2
= −β
3) Solve the matrix equation XA = (X + I)B where I is the identity matrix,

6

A=
4
−1

9 1
2 4 
0 −1

and
5

B=
2
−1

7 1
0 3.
1 2
4) Determine for what value of a ∈ IR the following matrix is regular and for those value find the inverse matrix

a+1
A= 0
0
0
a
−1

0
1 .
a
Linear Algebra-Lab 8
1) Determine if the following are linear transformations
1.1) f : IR2 → IR, f (x, y) = x + y,
1.2) f : IR2 → IR, f (x, y) = x + 1,
1.4) f : P 2 → IR2 , f (ax2 + bx + c) = (a + b, b + c),
1.3) f : IR2 → IR, f (x, y) = xy.
1.5) f : IR2 → P 1 , f (a, b) = b + a2 x.
2) Given the linear transformation l : IR3 → 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 : IR2 → IR2 , l(x, y) = (2x − 2y, −x + y), write the matrix associated to l
with respect to the standard basis of IR2 , find Ker l, Im l, its bases and dimensions. Find all vectors of IR2
that are mapped to (4, −2).
4) Given l : IR3 → IR3 , l(x1 , x2 , x3 ) = (x1 + 2x2 + 3x3 , 4x1 + 5x2 + 6x3 , x1 + x2 + x3 ), find Ker(l), Im(l), their
bases and dimensions.
5) Given l : IR3 → IR2 , l(x1 , x2 , x3 ) = (2x1 −x2 +3x3 , x1 +x2 +x3 ), find Ker(l), Im(l), their bases and dimensions.
4
3
6) Given the linear
 transformationl : IR → IR that has as associated matrix with respect to the standard
1 −1 1 1

bases A(l) = 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 : M2×2 → P 3 defined by l(A) = l
= a + (2a − b)x + (b + c)x2 + (a − b + c + d)x3 , find the
c d
matrix associated with l with respect to the standard bases. Is l an isomorphism?
Linear Algebra-Lab 9
1) In IR3 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 PB→C , PD→C , PB→D , PD→B .
2) Given l : IR2 → IR2 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 : IR3 → IR3 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 ∈ IR3 such that l(v) = (2, 3, 1).
4) Given the linear transformation lλ : IR3 → IR3 , lλ (x, y, z) = (2x+y, y −z, 2y +λz), write the matrix associated
to lλ with respect to the standard basis of IR3 , 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 lλ is an isomorphism?.
5) Given l : IR4 → IR3 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 ∈ IR4
such that Av = (13, 21, 5).
6) Given the transformation lh : IR3 → IR3 defined by lh (x, y, z) = (x − hz, x + y − hz, −hx + z), where h ∈ IR
is a parameter. Find, for all possible values of h, Ker(lh ), Im(lh ), their bases and dimensions.
Determine lh−1 (1, 0, 1) = {(x, y, z) ∈ IR3 : lh (x, y, z) = (1, 0, 1)}.
Linear Algebra-Lab 10
1) Given the linear transformations f : IR4 → IR3 , f (x, y, z, t) = (x − t, x + y, z + y) and g : IR3 → IR4 ,
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 : M2×2 → M2×2 by l(X) = M · X. Is
3 4
l an isomorphism? If possible, find its inverse.
3) Given the transformation l : IR3 → IR3 , 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
1

5) Determine if A = 3
6
−1
,B=
1

−3 3
−5 3 
−6 4
1
2
−1
1

2
and C =  0
0

and
−3

B = −7
−6
1
5
6
1
3
0

1
1  are diagonalizable.
2

−1
−1  are diagonalizable.
−2
Notice that the given matrices have the same characteristic polynomial but they are not similar.
6) Given the transformation l : IR3 → IR3 , 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 : IR2 → IR2 , 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 IR2 such that A(l,S,S)
is diagonal.
2) Given the linear transformation l : IR3 → IR3 , 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

−2
A= 1
−1
the following matrices (for each eigenvalue find a basis of the




1 0
2
1
1
0 1
−2 0 
B= 1
2
1 
C =  1 −1
1 1
−2 −2 −2
1 0
corresponding eigenspace)

4
1.
1
4) Consider the following system

−k
 kx + y − z =
(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 Ak associated to the system in 3), so that the system can be written in the form
  

x
−k
Ak  y  =  h + k 
z
2h + 1
Consider the linear transformation lk IR3 → IR3 associated to Ak with respect to the canonical basis of IR3 .
5.1) Determine for what values of k dim Im lk = 3.
5.2) Determine for what values of k dim Ker lk = 2.
5.3) For k = 0 is the transformation l0 diagonalizable?
Linear Algebra-Lab 12
1) Given the vectors v = (1, 5), u = (3, 4) in IR2 , with the standard inner product, find < u, v >, ||u||, ||v||.
2) In C[0, 1] with the standard inner product, i.e. < f, g >=
R1
f (t)g(t) dt, consider the functions f (t) = t + 2,
0
g(t) = 3t − 2,
3) Find cos θ =
h(t) = t2 − 2t − 3. Find < f, g >, < f, h >, ||f ||, ||g||.
<u,v>
||u||·||v|| ,
where θ is the angle between the vectors v, u, and
3.1) u, v ∈ IR4 , u = (1, 3, −5, 4), v = (2, −3, 4, 1),
9 8 7
1
3.2) u, v ∈ M2×3 , u = A =
, v=B=
6 5 4
4
2
5
3
.
6
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 IR4 .
6) Given, in IR5 , W = span{(1, 2, 3, −1, 2), (2, 4, 7, 2, −1)}, find a basis of the orthogonal complement W ⊥ .
7) In IR4 is given the space W = span{(1, 2, 3, 1)}, find an orthogonal basis of the orthogonal complement W ⊥ .
Linear Algebra-Lab 13
1) In IR4 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 IR4 . Find the coordinates of v = (a, b, c, d) with respect to S.
2) Find the Fourier coefficient c = <v,w>
||w||2 and the projection of v = (1, −2, 3, −4) along w = (1, 2, 1, 2) with
respect to the standard inner product in IR4 .
3) In IR4 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 .


2 1 1
4) Given the symmetric matrix A =  1 2 1  find an orthonormal real matrix P such that Pt 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 : IR3 → IR3 , 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 IR3 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 M1 = [−2, 2, 2], M2 = [0, 5, 3] and
M3 = [−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 √1662 .
12) 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
y−5
z
z+1
p: x−8
and q: x1 = y−1
5 = 1 = −1 ,
−2 = 1 .
14) Find the distance of the point Q = [3, 2, 1] from the plane containing the lines p and q, where
y−3
z−2
p: x+1
and q: X = [0, 4, 2] + t(1, 1, 0), t ∈ IR.
1 = 2 = 2 ,