University of Aarhus
Department of Mathematical Sciences
Calculus Quiz
week 36.1
Legend: A 4 indicates a correct response; a 8, indicates an incorrect response, in this case, the correct answer is marked with a l. Correct and follow
a 4,l for a solution. Return through n.
Fall 2004
2
Linear Mappings
1. Which map R2 → R is linear?
f (x, y) = xy
f (x, y) = x + y + 1
f (x, y) = xy + 1
f (x, y) = x − y
2. Which matrix is the associated matrix to the linear map
f (x, y, z) = (2x − y, y + z)?
1
0
0
1
2 −1
0 1
0
1
3. For composable linear maps f, g the matrix
M atr(g ◦ f ) = M atr(f )M atr(g)
Yes
No
2
0
0 −1
1 0
3
4. Which coefficient matrix of the system of linear equations?
1
0
0
1
2
2
x + y + z = 2, x − y + 2z = 0
1 1 1
1 1
1 −1 2
1 −1
1 2
2 0
5. For some system of 3 linear equations in 3 indeterminates the complete
set of solutions has exactly 3 elements.
Yes
No
6. The complete solution set to a homogeneous system of linear equations
in n indeterminates is a linear subspace of Rn ?
Yes
No
Solutions to Quizzes
4
Solutions to Quizzes
Solution to Quiz:
f (x, y) = xy is not linear:
f (2, 2) = 4 6= 2 = 2 · f (1, 1)
f (x, y) = x + y + 1 is not linear:
f (0, 0) = 1 6= 0
f (x, y) = xy + 1 is not linear:
f (0, 0) = 1 6= 0
f (x, y) = x − y is linear:
x
f (x, y) = x − y = 1 −1
y
See [LA] Theorem 4.
Solutions to Quizzes
5
Solution to Quiz:
For f (x, y, z) = (2x − y, y + z) the columns are
 
 
 
1
0
0
2
−1
0
f (0) =
, f (1) =
, f (0) =
0
1
1
0
0
1
giving
M atr(f ) =
2 −1
0 1
0
1
See [LA] Theorem 5.
Solutions to Quizzes
6
Solution to Quiz:
No, the correct formula is
M atr(g ◦ f ) = M atr(g)M atr(f )
“Beware the order of composition!”.
See [LA] Theorem 6.
Solutions to Quizzes
7
Solution to Quiz:
The rows of the coefficient matix are the coefficients of the respective indeterminates.
(1, 1, 1), (1, −1, 2)
giving
1 1 1
1 −1 2
See [LA] 5 Systems of linear equations.
Solutions to Quizzes
8
Solution to Quiz:
No, [LA] Theorem 8: The set of solutions is
u + NA
I.e. for any system:
• No solution
• Exactly 1 solution
• Infinitely many solutions
Solutions to Quizzes
Solution to Quiz:
Yes, this is [LA] Theorem 7.
9