Let A 13 0 0 0 13 0 0 0 13 and define TR3 rightarrow R3 by

Let A = [1/3 0 0 0 1/3 0 0 0 1/3], and define T:R^3 rightarrow R^3 by T(x) = Ax. Find T(u) and T (v) 0 0 when u = [3 6 -9], v = [a b c]. If A is a 6 times 5 matrix and T:R^P rightarrow R^q is given by T(x) = Ax, what values must p and q have? Consider the matrix A = [1 -3 5 -5 0 1 -3 5 2 -4 4 -4]. Find the set of all x belongs to R^4 that are mapped to the zero vector by the transformation x Ax. Let T: R^n rightarrow R^m be a linear transformation. Suppose that the set {u, v} is linearly independent but that {T(u), T(v)} is linearly dependent. Show that T(x) = 0 has a nontrivial solution. Find the matrix of each of the linear transformations T: R^2 rightarrow R^2. T rotates vectors about the origin through pi/4 radians counterclockwise. T first rotates vectors about the origin through pi/2 radians counterclockwise and then reflects points through the line x_2 = -x_1.

Solution

T(x+y)=A(x+y)....Definition of T

=Ax+Ay....Theorem MMDAA

=T(x)+T(y)Definition of T

and

T(x)=A(x)...Definition of T

=(Ax)....Theorem MMSMM

=T(x)...Definition of T

So by Definition LT, T is a linear transformation.

So the value of p and q will be order of matrix A.

That is