Linear Algebra And Its Applications 6th Edition Solutions Manual PDF guaranteed pass latest
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Course
MATH101
Institution
MATH101
Linear Algebra And Its
Applications 6th Edition
Solutions Manual PDF
guaranteed pass
latest
Question 1
Which of the following is a property of a vector space?
A) It must contain the zero vector.
B) It must be finite-dimensional.
C) It must contain an infinite number of vectors.
D) It...
,Linear Algebra And Its
Applications 6th Edition
Solutions Manual PDF
guaranteed pass
latest
Question 1
Which of the following is a property of a vector space?
A) It must contain the zero vector.
B) It must be finite-dimensional.
C) It must contain an infinite number of vectors.
D) It cannot be closed under scalar multiplication.
Correct Answer: A) It must contain the zero vector.
Rationale: A vector space must contain the zero vector as part of its axioms. It can
be finite or infinite-dimensional and must be closed under vector addition and
scalar multiplication.
Question 2
If AAA is a 3×33 \times 33×3 matrix, what is the maximum number of linearly
independent columns it can have?
A) 1
B) 2
C) 3
D) 4
,Correct Answer: C) 3
Rationale: A 3×33 \times 33×3 matrix can have at most 3 linearly independent
columns, corresponding to its number of rows. Thus, the rank of matrix AAA can
be at most 3.
Question 3
What is the determinant of the matrix A=(1234)A = \begin{pmatrix} 1 & 2 \\ 3
& 4 \end{pmatrix}A=(1324)?
A) -2
B) 2
C) 0
D) 1
Correct Answer: A) -2
Rationale: The determinant of a 2×22 \times 22×2 matrix A=(abcd)A =
\begin{pmatrix} a & b \\ c & d \end{pmatrix}A=(acbd) is calculated as ad−bcad -
bcad−bc.
For matrix AAA:
If the eigenvalue of a matrix AAA is λ\lambdaλ, what can be said about the
characteristic polynomial?
A) It is linear.
B) It is quadratic.
C) It has λ\lambdaλ as a root.
D) It is always positive.
Correct Answer: C) It has λ\lambdaλ as a root.
Rationale: The eigenvalue λ\lambdaλ of a matrix AAA is a solution to the
characteristic polynomial, which is given by det(A−λI)=0\text{det}(A - \lambda I)
= 0det(A−λI)=0. Thus, λ\lambdaλ is a root of this polynomial.
, Question 5
Which of the following statements is true about linear transformations?
A) They always increase the dimension of a vector space.
B) They map lines to lines or points.
C) They cannot be represented by matrices.
D) They are not defined for infinite-dimensional spaces.
Correct Answer: B) They map lines to lines or points.
Rationale: Linear transformations preserve the operations of vector addition and
scalar multiplication, meaning that they map lines in the domain to lines in the
codomain.
Question 6
Consider the vectors v1=(100)\mathbf{v_1} = \begin{pmatrix} 1 \\ 0 \\ 0
\end{pmatrix}v1=100 and v2=(010)\mathbf{v_2} = \begin{pmatrix} 0 \\ 1 \\ 0
\end{pmatrix}v2=010. Are these vectors linearly independent?
A) Yes
B) No
C) It depends on the context.
D) Only if the third vector is included.
Correct Answer: A) Yes
Rationale: Two vectors are linearly independent if the only solution to
c1v1+c2v2=0c_1\mathbf{v_1} + c_2\mathbf{v_2} = 0c1v1+c2v2=0 is
c1=c2=0c_1 = c_2 = 0c1=c2=0. Since v1\mathbf{v_1}v1 and v2\mathbf{v_2}v2
point in different directions, they are indeed linearly independent.
Question 7
What is the rank of the matrix B=(123000456)B = \begin{pmatrix} 1 & 2 & 3
\\ 0 & 0 & 0 \\ 4 & 5 & 6 \end{pmatrix}B=104205306?
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