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MT210 TEST 3 SAMPLE 1
ILKER S. YUCE
APRIL 19, 2011
QUESTION 1. THE PROPERTIES OF DETERMINANTS
Find the determinant of the matrix below. Specify whether the matrix has an inverse without trying to
compute the inverse.
ANSWER
Row reduce the given matrix to an upper triangular matrix and keep track of the operations you perform:
1 · R 1 + R 2 ↔ R 2 , 1 · R 1 + R 3 ↔ R 3 , −
1 2
· R 1 + R 4 ↔ R 4. Then perform 2 · R 2 + R 4 ↔ R 4 and R 2 ↔ R 3.
Note that there is only one switch. As a result, the determinant is -32 which is not 0. Therefore, the given
matrix is invertible.
QUESTION 2. CRAMER’S RULE, VOLUME, AND LINEAR TRANSFORMATIONS
Solve the linear system using Cramer’s Rule:
2 x 1 + 3 x 2 − x 3 = 2
3 x 1 − 2 x 2 + x 3 = − 1
− 5 x 1 − 4 x 2 + 2 x 3 = 3
ANSWER
x 1 =
x 2 =
x 3 =
The solution is ( − 5 / 11 , 36 / 11 , 76 / 11).
QUESTION 4. LINEARLY INDEPENDENT SETS; BASES
Let v ⃗ 1 = (1 , 1 , 1), v ⃗ 2 = (1 , 2 , 3) and v⃗ (^) 3 = (1 , 1 , 2).
a. Show that the vectors above are linearly independent.
ANSWER
There are three pivot positions and three vectors. Therefore, v⃗ (^) 1 ,⃗v (^) 2 v,⃗ (^) 3 are linearly independent.
b. Find the unique scalars(weights) c 2 , c 2 , c 3 such that v⃗ = (2 , 1 , 3) can be written as v ⃗ = cv 1 ⃗ 2 + cv 2 ⃗ 2 + cv 3 ⃗ 3.
ANSWER
Thus, we get c 1 = 0, c 2 = − 1 , and c 3 = 3 or v⃗ = 0 ·v⃗ (^) 1 + ( − 1) v ⃗ 2 + 3 v⃗ (^) 3.
QUESTION 5. RANK
Define a linear transformation T : P 2 Ï P 2 by T (p( x )) = p ′ ( x ).
a. Describe the range of T
ANSWER
Consider the vector space P 2 with the standard basis { 1 , x, x 2 }. Then, the transformation T is given as
T ( ax 2
- bx + c ) = 2 ax + b = 2 a ( x ) + b (1). In other words, a basis for the range of T is C = { 1 , x}.
Consequently,
Range of T = {s + tx : s, t ∈ R }.
b. Find dim ( R ( T )).
ANSWER
Since the basis C has 2 elements, dim ( R ( T )) = 2.
c. Find dim ( N ( T )).
ANSWER
By the RANK THEOREM, we have dim ( N ( T )) + dim ( R ( T )) = dim P 2 , i.e., dim ( N ( T )) = 3 − 2 = 1.
