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Is the following transformation linear
t : R x R > R x R : (x,y) > (2x  y, 0) 
Find the matrix of the following linear transformations relative to a natural basis.
t1: R x R > R x R : (x,y) > (2x  y, 0) t2: R x R > R x R : (x,y) > (2x  y, x) t3: R x R x R > R x R x R: (x,y,z) > (2x  y, 0, y +z) t4: R x R x R > R x R x R: (x,y,z) > (0, 0,y) 
Let t be a linear transformation of V. M is the set of all fixed points of t. Show that M is a subspace of V. 
Let V be the vector space with the complex numbers as vectors.
We choose the vectors 1 and i as a basis.
t is a linear transformation of V with matrix
[ 3 1 ] [ 4 3 ]Find the fixed points of t. 
Find the image of the vector (2,4) relative to the following linear transformations. Do this first without the matrix, and next with the matrix of t. t_{1} : R x R > R x R : (x,y) > (2x  y, 0) t_{2} : R x R > R x R : (x,y) > (2x  y, x) 
Take an origin O in the plane and orthonormal basis vectors e_{1} and e_{2}. A homothetic transformation h with center O and real factor k is a transformation of the plane such that h(v) = k v for all vectors v of that plane. Show that a homothetic transformation is a linear transformation and determine the matrix relative to the basis (e_{1}, e_{2}). Find the eigenvectors of h? 
V is the vector space R x R. We define a transformation
t of V such that the vector (x,y) is transformed in the vector (2x+3y,x2y).

Find the eigenvalues and the characteristic vectors of
t_{1} : R x R > R x R : (x,y) > (2x  y, 0) t_{2} : R x R > R x R : (x,y) > (2x  y, x) t_{3} : R x R x R > R x R x R: (x,y,z) > (2x  y, 0, y +z) t_{4} : R x R x R > R x R x R: (x,y,z) > (0, 0,y) 
Take an origin O in the plane and orthonormal basis vectors e_{1} and e_{2}.
The linear transformation t_{1} is the orthogonal reflection in the line y = x. Find the matrix and the eigenvectors of the transformation t. 
A linear transformation t has a matrix
[ 1m 2 4 ] [ 3 1 0 ] [ m m 2 ]Find the mvalues such that the nullspace of t is different from {0}. Find the nullspace corresponding with each of these mvalues. 
In V = R^{2} we take a natural basis B.
The set S = { (3 r, 7 r)  r in R and r not zero }
is a set of characteristic vectors of a linear transformation t.
In V we take a new basis B. The basisvectors are (3,1) and (2,1). Find the set of all the coordinates of the characteristic vectors in S relative to the new basis B. 
We start with two vectors v and w of a vector space V.
t is a linear transformation of V.
Show that: 
t is a linear transformation of a vector space V. u(1,1) is a fixed point of t. v(2,1) is in ker(t). Find the matrix of t. 
Take an origin O in the plane and orthonormal basis vectors e_{1} and e_{2}.
The linear transformation t has matrix [[3,1],[2,1]]. 
V is a vector space with basis e_{1} en e_{2}. t is a linear transformation of V.
t( u(1,3) ) = u'(5,8) t( v(2,1) ) = v'(3,5)Find the matrix of the linear transformation t relative to the given basis. 
In V = R^{2} we take a natural basis B.
The set S = { (3 r, 7 r)  r in R and r not zero }
is a set of characteristic vectors of a linear transformation t.
The linear transformation t' has matrix [[2,1],[4,1]] relative to the basis B. Find the set S' of all vectors v(x,y) such that t'(v) belongs to S. 
The rotation, with angle u radians (u not 0), about a fixed point o is a
linear transformation of the vector space of all vectors in a plane. Find the matrix of a rotation relative to an orthonormal basis (e_{1},e_{2}) in the plane Write this matrix for u = pi, and calculate the eigenvalues and the characteristic vectors. 
Let A = matrix of a linear transformation t. Prove that t has an eigenvalue 0 if and only if A is singular. 
The linear transformation t has, relative to an orthonormal basis
(e,u) , the matrix
[m m] [1 2]a) Calculate m such that (1,1) are the coordinates of a characteristic vector v. b) Calculate the coordinates of a characteristic vector w, linear independent of v and such that w is a unit vector. c) What is the matrix of t if we take v and w as a new basis in the vector space. d) Calculate the coordinates of e and u relative to this new basis. 
The vector v = (1,1) is a characteristic vector of the linear
transformation with matrix A =
[4 3] [7 8] a) What is the corresponding eigenvalue? b) Calculate 1998 A v 
Calculate all eigenvalues of the linear transformation t with matrix
[1 u v] [0 1 u] [0 0 1]Here, u and v are constant, non zero real numbers. Give for each eigenvalue the dimension of the associated vector space. 
Take an origin O in the plane and orthonormal basis vectors e_{1} and e_{2}. The linear transformation p is a nonorthogonal projection on a line through O. The image of u(4,5) by p is u'(2,1). Find the matrix of p. 
Let A = matrix of a linear transformation t and C is a regular matrix.
Prove that A and B = C^{1} .A.C have the same eigenvalues. 
A of a linear transformation t is a 2x2 matrix with two different eigenvalues. Show that the characteristic vectors corresponding with different eigenvalues are linear independent. Prove that, if you choose two linear independent characteristic vectors, as a new basis, the matrix of t is a diagonal matrix. 
The product of all eigenvalues of a matrix A is not 0. Show that A is a regular matrix. 
Take an origin O in the plane and orthonormal basis vectors e_{1} and e_{2}. The linear transformation t has as matrix [[2,2],[1,1]]. t transforms all points of the circle with equation x^{2}+y^{2} = 2 in points of another curve. Find the equation of this other curve. 
V is the real vector space of all polynomials in x with real coefficients and with a degree
lower than three. In V we take the basis B = (x^{2} , x , 1). We take a linear transformation t such that t(x^{2}) = x + m t(x) = (m  1)x t(1) = x^{2} + mFind :
