Algebra terminology of Calculus Questions

Q1. Prove Lemma 3.6.1 for columns.Q2.Verify that interchanging rows p and q
(q > p) can be accomplished using 2(q p) 1 adjacent
interchanges.
Q3. If u is a number and A is an n×n matrix,
prove that det(uA) = u
n det A by induction on n, using
only the definition of det A.
1
1.1
1.2
2
Q4.Answer, in your own words, and justifying yourself briefly: what can we say abo
ut the vector vk as k gets very large for a linear dynamical system with 3 × 3 matrix
A which has eigenvalues λ1 = λ2 = 1 with geometric multiplicity 2 and eigenvalue
λ3 = 1/3? (In other words, in the terminology of Calculus, what is the limit of vk as k g
oes to infinity, and why?)
Read the following two definitions carefully, and then answer the questions below.
Given a plane P, a line L and a vector v in R3 , we define the following new
plane and new line:
Definition
• P + {v} = {x + v | x ∈ P} and
• L + { v } = { x + v | x ∈ L }.
For example, if v = [0 0 1]t and L is the x-axis, then L + {v} is the line parallel to L
sitting one unit directly “above” L (in the z direction). In other words, the line L has
been “shifted” by the vector v.
This is a special case of much more general definition we could make: if A and B are any sets of vectors (say in R3 ) or numbers or
matrices, or really anything else that we can ’add’ then we can let A + B be the set {a + b | a ∈ A, b ∈ B}.
Definition
We say that a plane P or line L is original if it goes through the origin.
Now, if P is an original plane, we define a plane P ′ to be a a shift of P if
P ′ = P + {v} for some non-zero v. In this case, we say that we shifted P by v
to get P ′ or that P ′ is P shifted by v.
(We can make analogous definitions for lines: e.g. if L is an original line, we
define a line L ′ to be a shift of L if L ′ = L + {v} for some non-zero v.)
2.1
2.2
2.3
Explain why if P is an original plane, and w and x are in P, w + x is also in P. (i.e.
the sum of two vectors in an original plane is always a vector in that same plane.)
Explain why the following statement is false: “If P ′ is a shift of some (original)
plane P in R3 , say P ′ = P + {v}, then for any w and x in P ′ , w + x is in P ′ .” (This
shows that we can’t simply add two vectors in a shift of a plane and have the result be a
vector in that shifted plane.)
Suppose we “fix” the statement in the the previous part by changing it to the
following: “If P ′ is a shift of some (original) plane P in R3 , say P ′ = P + {v}, then
for any w and x in P ′ , w + x − v is in P ′ .” Explain why this statement is true.
Hint: Is there a way to add the vectors “inside P” and then shift the result to P ′ ?
Alternately, it could help to think of V (the point for v) as the “origin” for P ′ .
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