Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f:R² → R² defined by f(r, y) = (r + y. 1). f : R³ → R defined by f(r, y, z) = 2r – 3y + 4z. f:R? +R defined by f(r, y) = ry.
Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f:R² → R² defined by f(r, y) = (r + y. 1). f : R³ → R defined by f(r, y, z) = 2r – 3y + 4z. f:R? +R defined by f(r, y) = ry.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Math 125
Exercise - 3
Linear Transformation:
1. Show that the following maping f are linear:
f: R - R² defined by f(x, y) = (x + y, x).
f: R* +R defined by f(r, y, z) = 2.r – 3y + 4z.
• f : R? → R defined by f(r, y) = ry.
• f : R² → R³ defined by f(x, y) = (r +1, 2y, r + y).
f: R* - R² defined by f(x, y, z) = (|r|, 0).
Linear Combination and Span of Vector Space and Sub Space:
2. Show that the ry plane W = (x, y,0) in R* is generated by:
(i)u =
and
(ii)u =
and v =
3. Show that that the vector u =
-5 is a linear combination of the vectors v, =
3
2
-4 and v3 =
-5
4. For which value of k the vector u =
-2
is a linear combination of the vectors
3
and ty =
of the vectors vi =
and tz =
Linearly In-dependency and Dependency
5. Determine whether or not the following vectors i R are linearly dependent:
(1, -2, 1), (2, 1, –1), (7, -4, 1)
(1, 2, –3), (1, –3, 2), (2, –1,5)
(1, –3, 7), (2,0, –6), (3, –1, –1), (2,4, –5)
(2, –3, 7), (0,0,0),, (3. -1, -4)
![Math 125
Exercise - 4
1. Determine whether or not the following vectors form a basis for the vector space R³:
• (1, 1, 1) and (1, –1, 5).
• (1, 2, 3), (1,0, -1), (3, –1,0) and (2, 1, –2).
• (1, 1, 1), (1, 2, 3) and (2, –1, 1).
• (1, 1, 2), (1,2, 5) and (5, 3, 4).
2. Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) and
(3, 8, –3, –5). find the basis and the dimension of W and the extend the basis of W
to a basis of the whole space R.
3. Find the basis of columnspace and nullaspace of matrices:
Г1 3 1
-2
-3]
2
-31
1
3
1 4
A =
3
-1
2
-2
-4
B =
1.
2 3 -4 -7 -3
and C =
3
-2 -1
5
-1
3 8
1
-7
-8
4
-2
and also find the rank and nullity of the matrices.
4. Find the Eigenvalues of the matrices:
ГО -3 31
A = 3 -5 3 B =
-3 1
-17
-7 5 -1
6.
-6 4
-6 6
Also find the basis for eigen-space. Are the matrices diagonalizeable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95815611-eaad-4858-ad37-e304c653cccc%2F213fcfff-2f90-4326-a90b-4438b42a4ed9%2F650na55_processed.png&w=3840&q=75)
Transcribed Image Text:Math 125
Exercise - 4
1. Determine whether or not the following vectors form a basis for the vector space R³:
• (1, 1, 1) and (1, –1, 5).
• (1, 2, 3), (1,0, -1), (3, –1,0) and (2, 1, –2).
• (1, 1, 1), (1, 2, 3) and (2, –1, 1).
• (1, 1, 2), (1,2, 5) and (5, 3, 4).
2. Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) and
(3, 8, –3, –5). find the basis and the dimension of W and the extend the basis of W
to a basis of the whole space R.
3. Find the basis of columnspace and nullaspace of matrices:
Г1 3 1
-2
-3]
2
-31
1
3
1 4
A =
3
-1
2
-2
-4
B =
1.
2 3 -4 -7 -3
and C =
3
-2 -1
5
-1
3 8
1
-7
-8
4
-2
and also find the rank and nullity of the matrices.
4. Find the Eigenvalues of the matrices:
ГО -3 31
A = 3 -5 3 B =
-3 1
-17
-7 5 -1
6.
-6 4
-6 6
Also find the basis for eigen-space. Are the matrices diagonalizeable?
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