Linear Algebra - Fall 2023
Exam 2
1. (a) Find a basis in ker(A), where
(b) Find a basis in Im(A), where
2. Let A be the matrix
(a) Find the rank and nullity of this matrix.
(b) Find a matrix C whose image is the kernel of the linear transformation whose matrix is A.
3. Let P2 = {a + bx + cx2|a, b, c 2 R} be the vector space of two polynomial functions of degree less or equal to two. Consider the map T : P2 ! P2 given by
T(f) = f + f' + f''
where f' represents the derivative of the polynomial f.
(a) Verify, using the definition (or otherwise) that T is a linear transformation.
(b) Find the matrix A of the linear transformation T with respect to the basis U = (1, x, x2).
PART II of 3.
(c) Find the change of basis matrix S from B to U, where B is the basis B = (1,(x + 1),(x + 1)2) (you don’t have to verify here that B is a basis).
(d) Using any method, find the matrix B of the linear transformation T with respect to the basis B.
4. Find an orthonormal basis in the subspace V of R4, where
5. Let v1, v2 2 R4 be the vectors
(a) Check that (v1, v2) is an orthonormal basis in V = Span{v1, v2}.
(b) Find the projection projV (w) of the vector w = onto V .
PART II of 5.
(c) What are the coordinates of the vector z = projV (w) with respect to the basis (v1, v2)?
(d) Determine the matrix of the linear transformation T(x) = projV (x).
6. A 4 x 3 matrix has rank 2. Answer the following:
(a) What is the nullity of this matrix?
(b) Are the columns of A linearly independent?
(c) If the system Ax = b is consistent, how many free variables does it have, if any?
(d) What is dim(ker(A)) and dim(Im(A))?
BONUS A+
Let A be a 2 x 3 matrix and B be a 3 ⇥ 2 matrix, such that AB = I2, the 2 ⇥ 2 identity matrix. Consider the linear transformation T : R3 ! R3 defined by
T(x) = BAx
Show that T is a projection onto a plane V and describe V in terms of A and B.