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