Academic year Mathematics I
Linear Maps
1. Calculate the linear maps defined by the following matrices
0 0 0 1 2 0 1
0 0 0
a) 0 1 0 , b) 1 −1 2 , c) , d) 4 , e) (1 2 3)
3 4 4
1 0 0 5 4 0 −3
Solution: (a) f : R3 −→ R3 , f (x1 , x2 , x3 ) = (0, x2 , x1 );
(c) f : R3 −→ R2 , f (x1 , x2 , x3 ) = (0, 3x1 + 4x2 + 4x3 );
(d) f : R −→ R3 , f (x) = (x, 4x, −3x);
2. Given a linear map f : R3 −→ R3 such that f (1, 0, 0) = (2, 4, 0), f (0, 1, 0) = (−1, 4, 1) y
f (0, 0, 1) = (5, −2, 1). Calculate the associated matrix, the kernel and the image of f .
Solution: Kerf = {(0, 0, 0)} e Imf = R3 .
3. Consider the linear map
f : R3 −→ R3 , f (x, y, z) = ( x + 2y , 2x + 5y − z , y − z ).
a) Calculate the matrix of the linear map.
b) Calculate a basis of the kernel and the implicit equations of the image of f .
4. Consider the linear map f : R3 −→ R3 definida por
f (x, y, z) = (x + y, y + 3z, x − 3z).
a) Calculate the matrix of the linear map.
b) Calculate a basis of Kerf and the implicit equations of Imf .
5. Consider the linear map
f : R2 −→ R3 , f (x, y) = (x + 2y, −x + y, −3x).
a) Calculate the matrix of the linear map.
b) Calculate a basis of the kernel and the implicit equations of the image of f .
c) Belongs the vector (1, 1, 1) to the image of f ?
6. Consider the linear maps
f : R3 −→ R4 , f (x, y, z) = ( 3x + y , 0 , x − y + z , 4y − 3z ),
g : R4 −→ R3 , g(x, y, z, t) = ( x + y + z − t , 2x + y + 2t , y − 4t )
h : R3 −→ R3 , h(x1 , x2 , x3 ) = (x1 + x2 + x3 , 2x1 + 3x2 + x3 , 2x1 + x2 + 3x3 )
j : R3 −→ R2 , j(x, y, z) = (x + y − z, 2x + y)
a) Calculate the matrix of each linear map.
b) Calculate the kernel and the image of f y g.
UPV. Departamento de Matemática Aplicada 1
Academic year Mathematics I
7. Consider the linear map f : R3 −→ R3 whose associated matrix is
1 2 0
A= 0 1 1
1 3 1
a) Calculate a basis of Kerf and the implicit equations of Imf .
b) Calculate a to ensure that (2, a, 1) belongs to Kerf .
8. Consider the linear map f : R3 −→ R4 with associated matrix
1 2 1
2 5 1
A= 1
.
1 2
1 0 3
a) Calculate the equations of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) (1, 2, 1, 1) ∈ Imf ?
9. Consider the linear map f : R3 −→ R4 given by
f (x, y, z) = ( x − y + z , −x + z , y , x + 2z )
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) (1, 0, 1, 3) ∈ Imf ?
10. Consider the linear map f : R4 −→ R3 given by
f (x1 , x2 , x3 , x4 ) = ( x1 − 2x2 + x4 , 2x1 + x2 + 5x3 + 7x4 , −x1 + 3x2 + x3 )
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) (1, 0, 3) ∈ Imf ?
11. Consider the linear map f : R3 −→ R2 given by
f (x, y, z) = ( x + y + 2z , 3x + 3y + 6z )
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
UPV. Departamento de Matemática Aplicada 2
Academic year Mathematics I
c) Calculate implicit equations of Imf .
d ) Calculate the value of a such that (a, 1, −1) ∈ Kerf
12. Consider the linear map f : R3 −→ R3 given by
f (x, y, z) = ( x , y + 2z, −x )
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) Calculate the value of a such that (a, 1, −1) ∈ Imf
1 −2 1
0 1 −1
13. Consider the linear map f : R3 −→ R4 with matrix A = .
1 1 −2
−1 0 1
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) Calculate the value of a and b such that the vector (a, b, 3) ∈ Kerf
14. Consider the linear map f : R3 −→ R3 given by
f (x1 , x2 , x3 ) = ( x1 + 2x2 + x3 , 2x1 + x2 + 5x3 , x1 + x2 + 2x3 ).
a) Calculate the matrix of f .
b) Calculate a basis of Kerf .
c) Calculate implicit equations of Imf .
d ) Calculate the value of a, b, c such that (a, −2, b) ∈ Kerf and (1, 2, c) ∈ Imf .
UPV. Departamento de Matemática Aplicada 3
