Determine whether the members of the given set of vectors are
linearly independent. If they are linearly dependent, find a linear
relation among them of the form c1x(1) + c2x(2) + c3x(3) = 0. (Give
c1, c2, and c3 as real numbers. If the vectors are linearly
independent, enter INDEPENDENT.) x(1) = 9 1 0 , x(2) = 0 1 0 , x(3)
= −1 9 0
For each family of vectors, determine wether the vectors are
linearly independent or not, and in case they are linearly
dependent, find a linear relation between them.
a) x1 = (2, 2, 0), x2 = (0, 2, 2), x3 = (1, 0, 1)
b) x1 = (2, 1, 0), x2 = (0, 1, 0), x3 = (1, 2, 0)
c) x1 = (1, 1, 0, 0), x2 = (0, 1, 1, 0), x3 = (0, 0, 1, 1), x4 =...
Determine all values of a for which the following set of vectors
is dependent or independent. You can select 'always', 'never', 'a =
', or 'a ≠', then specify a value or comma-separated list of
values. ⎧ ⎪ ⎨ ⎪ ⎩⎫ ⎪ ⎬ ⎪ ⎭a001, 12−2−2, 48−8−9Dependent: When a =
0Independent: When a ≠ 0
A basis of a vector space V is a maximal linearly independent
set of vectors in V . Similarly, one can view it as a minimal
spanning set of vectors in V . Prove that any set S ⊆ V spanning a
finite-dimensional vector space V contains a basis of V .
Prove the follwing statements
Suppose that S is a linearly independent set of vectors in the
vector space V and let w be a vector of V that is
not in S. Then the set obtained from S by adding w
to S is linearly independent in V.
If U is a subspace of a vector space V and dim(U)=dim(V), then
U=V.
In each of the following situations, identify the independent
variable and dependent variable.
Determine whether the growth is linear or exponential and explain
why by the definition of linear and
exponential growth (Do not present the table)
a) The price of gasoline has been rising 2% per month.
b) The total cost of a taxi ride increases by $ 5 per 2
miles.
c) The population of a bacteria is double every 20 hours.
Determine whether the following pairs of functions are linearly
independent or not on the whole real line.
Determine whether the following pairs of functions are linearly independent or not on the whole real line. BEWARE: You only get 3 tries. Linearly dependent * 1. f(theta) = 14 cos 3theta and g(theta) = 56cos^3 theta - 42 cos theta. Linearly dependent 2. f(t) = t^2 + 14t and g(t) = t ^2 - 14t Linearly dependent 3.f(t)=t and g(t)=|t|