1. Given a 3x4 matrix A and it is known that rank(A) = 2. The following sentences are always TRUE: a. Dimension of Null(A) is 1 b. Dimension of Row(A) is 1 C. The kernel of the function f: x → Ax has dimension 2 d. (Null(AT))= {0}
1. Given a 3x4 matrix A and it is known that rank(A) = 2. The following sentences are always TRUE: a. Dimension of Null(A) is 1 b. Dimension of Row(A) is 1 C. The kernel of the function f: x → Ax has dimension 2 d. (Null(AT))= {0}
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.2: Length And Angle: The Dot Product
Problem 17EQ
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![Multiple choice.
Choose the correct answer without giving a reason
1. Given a 3x4 matrix A and it is known that rank(A) = 2. The following sentences are always TRUE:
a. Dimension of Null(A) is 1
b.
Dimension of Row(A) is 1
C. The kernel of the function f: x → Ax has dimension 2
d.
(Null(AT)) = {0}
2. A is the set containing the vectors in the finite dimensional vector space V. Which of the
following sentences is always true?
a. If A is linearly independent, then is orthogonal
b.
If A is orthogonal, then A is linearly independent
C. If A is orthonormal, then A is linearly independent
d. If A is orthonormal, then A base V
3. A is a square matrix of size nxn. The following sentences are equivalent to λ are the eigenvalues
of A, EXCEPT....
a.
There is a nonzero vector x such that Ax =λx
b. SPL Ax = x has infinitely many solutions
C.
det (λI - A)² > 0
d. SPL (A-AI)x = 0 has a nontrivial solution
e. det(A) = det(λ)
4. The following are the properties of the linear operator T in R" which is one-to-one (injective)
EXCEPT.......
a. T onto (surjective)
b. [T]¹=[T¹]
c. T isomorphism
d. Nullity(T) = n
e. The set of rows [T] is the basis of Rn](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780135a8-f701-479c-be02-38c3a3de7347%2F777625da-b4e4-43f0-8f22-b3d6016507fc%2F7fqciom_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Multiple choice.
Choose the correct answer without giving a reason
1. Given a 3x4 matrix A and it is known that rank(A) = 2. The following sentences are always TRUE:
a. Dimension of Null(A) is 1
b.
Dimension of Row(A) is 1
C. The kernel of the function f: x → Ax has dimension 2
d.
(Null(AT)) = {0}
2. A is the set containing the vectors in the finite dimensional vector space V. Which of the
following sentences is always true?
a. If A is linearly independent, then is orthogonal
b.
If A is orthogonal, then A is linearly independent
C. If A is orthonormal, then A is linearly independent
d. If A is orthonormal, then A base V
3. A is a square matrix of size nxn. The following sentences are equivalent to λ are the eigenvalues
of A, EXCEPT....
a.
There is a nonzero vector x such that Ax =λx
b. SPL Ax = x has infinitely many solutions
C.
det (λI - A)² > 0
d. SPL (A-AI)x = 0 has a nontrivial solution
e. det(A) = det(λ)
4. The following are the properties of the linear operator T in R" which is one-to-one (injective)
EXCEPT.......
a. T onto (surjective)
b. [T]¹=[T¹]
c. T isomorphism
d. Nullity(T) = n
e. The set of rows [T] is the basis of Rn
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