Linear Algebra (Part 3)

Problem 1

Suppose that V and W are subspaces of  (ℝ^12) and dim(V)=10 and dim(W)=8.  Find the possible values of dim(V⋂W).

Problem 2

Suppose that T is a linear transformation from (ℝ^2) to (ℝ^2) and T((({{1}, {3}})))=(({{2}, {7}})) and T((({{2}, {1}})))=(({{-1}, {1}})).  Find the value of T((({{3}, {5}})))

Problem 3

Show that the function T(f(x))=x f'(x)+f(x) is a linear transformation from (Subscript[P, 2]) to (Subscript[P, 2]).  Find its kernel, its nullity, and the dimension of its range.

Problem 4

Suppose that T is the linear transformation from (ℝ^3) to (ℝ^2) given by T((x,y,z)=(x+y,y+z).  Find the kernel of T, the nullity of T, and the matrix for T (with respect to the “standard bases” for  (ℝ^3) and  (ℝ^2)).

Problem 5

For the linear transformation T((x,y))=(x+2y,3x+2y):
	a) Find the matrix A which represents it with respect to the “standard basis” for (ℝ^2).
	b) Find a basis B for (ℝ^2) which consists of eigenvectors for A.
	c) Find the matrix for T with respect to the basis B.

Problem 6

Let (OverVector[u])=<2,1,0,2> and (OverVector[v])=<3,0,3,2>, find ||(OverVector[u])||, ||(OverVector[v])||, and the sine of the angle between then vectors.

Problem 7

For the inner product space for continuous functions on [-π,π], find ||sin(x)||.  Show that sin(x) and cos(x) are orthogonal in that space.

Problem 8

Find the projection of the vector ||(OverVector[u])||=<2,1,3,-4> onto the vector ||(OverVector[v])||=<2,1,-1,1>.  If (OverVector[p]) is the projection, what is the angle between (OverVector[v]) and (OverVector[u])-(OverVector[p])?

Problem 9

In the space of continuous functions on [0,1], find the projection of (x^2) onto x+1.