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Math 308 Practice Final Exam

Problem 1: Solving a linear equation

Given matrix

and vector

(a) Solve Ax = y (if the equation is consistent ) and write the general solution x in
(vector) parametric form.

Write a basis for the null space of A. Basis = ___________________

(b) What is the dimension of the range of A? Dimension = _____

(c) Is y in the span of the row vectors of A? Yes? ___ No? ___

Problem 2: Conclusions from echelon form.

In each case, we start with a matrix A and vector and tell what one will get by reducing
the augmented matrix of the system Ax = y to echelon form. Answer the questions in
each case using this information. It is possible some equations will have no solution.

A y Echelon form of augmented
matrix of Ax = y.
(a) Write the general solution for Ax = y in (vector) parametric form
Solution:

(b) What is the dimension of the null space of A? Dimension = ____

(c) Write down a basis for the null space of A. Basis =
(d) Is y in the range of A? Yes? __ No? __
(e) What is the dimension of the range of A? Dimension = ______
(f) Write down a basis of the range of A. Basis =
(g) Are the rows of A independent? Yes? __ No? ___
(h) What is the dimension of the row space of A? Dimension = ____

B z Echelon form of augmented matrix of Bx = z.
(a) Write the general solution for Bx = z in (vector) parametric form
Solution:
(b) What is the dimension of the null space of B? Dimension = ______
(c) Write down a basis for the null space of B. Basis =
(d) Is z in the range of B? Yes? No?
(e) What is the dimension of the range of B? Dimension = ______
(f) Write down a basis of the range of B. Basis =
(g) Are the columns of B independent? Yes? No?
C Reduced row echelon form of C.
(a) Is C invertible? Yes? ___ No? ___
(b) Are the columns of C independent? Yes? ___ No? ___
(c) Write down a basis for the null space of C.

Problem 3: Compute matrix products and inverses

Compute the stated matrix products (if defined) for these matrices.

Compute each of the following matrix products or other matrices (if defined):

A-1 C-1
AB BA
CD BC
CDT CTD

Problem 4: Find the eigenvalues and eigenvectors

Find the eigenvalues and eigenvectors of matrix

If possible, diagonalize F, i.e., write F = UDV, where D is diagonal.

U =_____________ D = ________ V =_______________

Problem 5: Given the eigenvalues find the eigenvectors

Given that 1 and -3 are the eigenvalues of the matrix
find the

eigenvectors of this matrix. Hint: First find whether there is a third eigenvalue. If you
compute the determinant of C, knowing 2 eigenvalues, you can find the third root of the
characteristic
polynomial without computing det(C - λI).

If possible, diagonalize C, i.e., write C = MDN, where D is diagonal. You DO NOT
need to compute the inverse of any matrix in this problem. If a matrix is the inverse of a
known matrix, just write it as the inverse.

M =________ D =______________ N =__________________

Problem 6: Compute orthogonal projections

(a) Compute m = the projection of h on Span({u}). (The formula should be computed
numerically , but you need not simplify fractions , etc., in your answer.)

(b) Compute g = the projection of h on Span({u, v}). (The formula should be computed
numerically, but you need not simplify fractions , etc., in your answer.)

Problem 7: Matrix of Linear Transformation

(a) If T is the linear transformation of R2 that rotates the plane by an angle so that the
point (1, 0) is rotated to (3/5, 4/5). What is the matrix A of this transformation?

(b) Is the matrix A an orthogonal matrix? Yes? ___ No? ___

(c) Is the matrix 2A an orthogonal matrix? Yes? ___ No? ___
Show why in both cases.
(d) Given the vector if x is a vector in R3, then let T(x) = (u ∙ x)u. Is T a linear
transformation? If so, what is its matrix (with respect to the standard basis).

Problem 8: Least squares solution

You have the following data for variables x , y, z

x y z
0 0 3
1 0 4
0 1 4
1 1 1

(a) Now suppose that you want to fit this data to a function z = a + bx + cy, where, a, b, c
are some constants that you need to solve for. Write a system of equations for
variables a, b, c that can be solved if there is an exact fit of the function to the data.

(b) Find the lease-squares "solution" a, b, c to this system and check the values of z = a +
bx + cy at the data points to see how close a fit it is.

Least squares solution : z = ___ + ___ x + ___ y

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