Topic
Q:
Do vectors
\( v_{1}=(-8,2,-3,2), v_{2}=(4,1,-5,-4), v_{3}=(-3,-1,4,3), v_{4}=(2,-1,0,-1) \)
form a basis of \( R^{4} \) ? Answer this question by using
Q:
(Hint: verify that \( \left.p_{A}(x)=(x-1)^{2}(x+2)(x+1)\right) \)
Let \( a_{1}=(2,4,-6,8), a_{2}=(-1,-3,5,7), a_{3}=(3,9,-4,6), a_{4}=(-2,-8,3,9) \).
For the subset \( S=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\} \) of \( \mathbb{R}^{4} \) determine whether \( S \) is linearly
independent or not. If \( S \) is linearly dependent, show how to express one
vector in \( S \) as a linear combination of the others.
Q:
Linear Algebra
Diagonalize (find characteristic polynomial, eigenvalues, eigenvectors, ma-
trix \( P \) and a diagonal matrix \( D \) such that \( A=P D P^{-1} \). Find \( P^{-1} \) and
verify that the product \( P D P^{-1} \) is equal to \( A \) ) the following matrix
\[ A=\left(\begin{array}{cccc}0 & 0 & 2 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1\end{array}\right) \]
Q:
a) Determine the rank of: \( A=\left(\begin{array}{cccc}1 & 2 & 0 & -1 \\ 2 & 6 & -3 & -3 \\ 3 & 10 & -6 & -5\end{array}\right) \).
Q:
1. Example Using Gauss-Jordan method determine the inverse of the ma-
trix
\[ A=\left[\begin{array}{ccc}5 & 2 & -3 \\ 3 & 1 & -2 \\ 2 & -3 & -4\end{array}\right] \in \mathbb{R}^{3 \times 3} \]
Q:
d) Explain the three (3) basic operating modes of an emergency lighting system.
Q:
Si puo' ridurre l'effetto degli errori
accidentali effettuando piu' misurazioni
della stessa grandezza e calcolandone
poi il valor medio.
Vero
Falso
Q:
Exercises
1. Which of the following sentences are propositions? For those that are, indicate
value.
a. 123 is a prime number.
b. 0 is an even number.
c. \( x^{2}-4=0 \).
d. Multiply \( 5 x+2 \) by 3 .
e. What an impossible question!
2. State the negation of each of the following statements.
a. \( \sqrt{2} \) is a rational number.
b. 0 is not a negative integer.
c. 111 is a prime number.
Q:
Sonia wanted to make ice pops. She filled cups with 150
mL of juice and put them in the freezer.
How will freezing the juice affect its mass?
a Freezing the juice will not change its
mass.
b Freezing the juice will decrease its mass.
c Freezing the juice will first increase and
then decrease its mass.
Q:
What would cause water to turn from a gas to a liquid?
a decreasing the water's temperature
b increasing the water's temperature
d increasing the water's mass of the above
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