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This is an exam for the ma 242 - linear algebra course, covering topics such as systems of equations, linear independence, linear transformations, span, matrix products, and true or false questions related to matrix properties. It includes 7 questions, with a total score of 116 points.
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MA 242 โ Linear Algebra
Exam #
Instructions: To receive full credit you must show all work. Explain your
answers fully and clearly. You may refer to theorems/facts in the book or from
class. No calculators, books or notes of any form are allowed. Good luck!
Question Score Out of
Total 116
Consider the system of equations
x 1 + 3x 2 โ 5 x 3 = 4
x 1 + 4x 2 โ 8 x 3 = 7
โ 3 x 1 โ 7 x 2 + 9x 3 = โ 6.
Find the general solution to these equations in parametric form. What
geometric shape does the solution space form?
(a) Define what it means for T : R
n โโ R
m to be a linear transforma-
tion.
(b) Let T : R 2 โโ R 2 be defined by T (x 1 , x 2 ) = (x 1 +x 2 , x 1 ). Determine
the matrix associated to T.
(c) Let T : R
2 โโ R
2 be the linear transformation defined by first
rotating by ฯ/2 radians counterclockwise and then reflecting over
the x-axis. Determine the matrix associated to T.
(b) How many solutions (if any) does the equation(a) Ax = b have where
and b =
Explain your answer.
(a) Does there exist a map from R
4 to R
7 that is both one-to-one and
onto? If yes, give an example. If no, explain why not.
(b) Does there exist a map from R
3 to R
2 that is onto, but not one-to-
one? If yes, give an example. If no, explain why not.
Consider the matrices
and B =
Only one of the products AB and BA makes sense. Determine which one
and compute that product.
Correct answers score 4 points each. Incorrect answers score 0
points. Leaving a question blank scores 2 points.
(a) Every matrix is row equivalent to a unique matrix in echelon form.
(b) It is possible for the equation Ax = b to have no solutions while
at the same time for the equation Ax = 0 to have infinitely many
solutions.
(c) If the columns of a 5 by 4 matrix are linear independent then the
columns span R
5 .
(d) If n < m and T : R
n โโ R
m is a linear transformation, then T is
one-to-one.
(e) If b is in the span of the columns of A, then the matrix equation
Ax = b has at least one solution.
(f) If A is matrix with a pivot point in every row and B is a matrix with
a pivot point in every row, then the product AB will have a pivot
point in every row.
(g) If A contains a row of all zeroes, then the equation Ax = 0 has no
solutions.
(h) A homogenous system of three equations with seven unknowns will
always have infinitely many solutions.
