[M] In Exercises 30 and 31, find the B-matrix for the transforma- tion x+ Ax where B = {b1, b2, b3}. %3D 6 -2 -2 1 -2 2 -2 30. A = 3 bị = b2 ,b3 %3D
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![is {1, t, t2,t} for
the eigenvalues of A. Verify this statement for the case when
A is diagonalizable.
e transformation
27. Let V be R" with a basis B = {b1, ..., b,}; let W be R"
with the standard basis, denoted here by ɛ; and consider the
identity transformation I : R" → R", where I(x) = x. Find
the matrix for I relative to B and E. What was this matrix
called in Section 4.4?
28. Let V be a vector space with a basis B = {b,..., b,}, let W
be the same space V with a basis C = {c...., c,}, and let I
be the identity transformation I : V → W. Find the matrix
for I relative to B and C. What was this matrix called in
Section 4.7?
= Ax. Find a
nal.
29. Let V be a vector space with a basis B = {bj,..., b,}. Find
the B-matrix for the identity transformation I: V →V.
[M] In Exercises 30 and 31, find the B-matrix for the transforma-
tion x→ Ax where B = {b1, b2, b3}.
b1 =
Г6 -2 -2
30. А — | 3 1 -2 |,
|2 -2
Ax.
hat A is not
b1 =
1
, b2 =
1, b3 =
3
is a 3 x 3
-7 -48 -16
31. A =
1
14
6.
exist a basis
nal matrix?
-3 -45 -19
-3
-2
1, b2 =
b1 =
-3
b3
are square.
= invertible
3 for some
32. [M] LetT be the transformation whose standard matrix is
given below. Find a basis for R with the property that [ T ]R
is diagonal.
en find an
-6
9.
-3
A =
-1 -2
1
is similar
-4
4
0.
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