Vectors and Matrices : Operations and Transformations

1) Let u=(2,3,0), and v=(-1,2,-2). Find

a) ||u + v||
b) ||u|| + || v ||
c) Find two vectors in R³ with norm 1 orthogonal to be both u and v
d) Find norm of vector u / || u ||

2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal?

3)

a) Find the standard matrix [T] for the linear transformation T: R^4 -> R³ defined by the formula: T(x1, x2, x3, x4) = ( x1- 2x3+ x4, -x3, x2 + x4)

b) Use the standard matrix [T] to find T(x) where x = ( 0, 1, -3, 2)

4)
Consider two linear operations T1 and T2 on R² given by the standard matrices
[T1] = -1 0 [T2] = 2 0
0 1 0 2
a) Give geometrical interpretation of the action of each transformation T1 and T2.

b) Determine whether T1 and T2 are one-to-one operators.

c) Find the standard matrices for compositions of linear transformations T1 º T2 and T2 º T1

d) For the vector v = (1,2) find (T1 º T2) (v) and (T2 º T1) (v)

e) Give a geometrical illustration for T1 º T2 and T2 º T1

Solution Summary

Operations and transformations of vectors and matrices are investigated. The solution is detailed and well presented.

... The operations to transform a system ... does not deal with the actual systems in terms of equations but instead makes use of the coefficient matrix (which is ...

... 2. Find the images of the basis vectors under T ... the eigenvectors, we ﬁnd solutions to the matrix equation (T ... the set of all multiples of the vector x, that is ...

... The vector algebra mathematics is also. ...Matrices are. ... programs use a technique to draw linear vectors using the "pen up" and "pen down". method. ...

... its eigenspaces is equal to n. Since matrices and linear ... (Note that there should be n such vectors since they ... of dimension n). Each is a column vector, and we ...

... The third rotation is by γ radians about the x-axis, and its matrix is: ... Now, we just apply these rotation matrices in the proper order to a vector: ...

... 3. Draw vectors u and v tail-to-tail so that ... to-tail, then find components for the vector that results ... what must be true of the dimensions of these matrices? ...

... zero v) The 3×3 matrices A such that vector [1; 2 ... i) The space of all 2X2 matrices A such that A[1 ... basis is given, for instance, by points and .The matrix of in ...

... of the elements on the diagonal) is invariant under transformations of the basis vectors. ... So, you just have to apply the matrix A to the state vector |x(t ...

... Note that Λµν and ηµν correspond to the matrices L and ... They are called contravariant vectors. ... the diﬀerentials dxµ and dτ are tensors (a vector and a ...