Linear Algebra And Its Applications 6th Edition Solutions Manual PDF guaranteed pass latest

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: det(A)=(1)(4)−(2)(3)=4−6=−2.text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = - (A)=(1)(4)−(2)(3)=4−6=−2. Question 4 If the eigenvalue of a matrix AAA is λlambdaλ, what can be said about the characteristic polynomial? A) It is linear. B) It is quadratic. C) It has λlambdaλ as a root. D) It is always positive. Correct Answer: C) It has λlambdaλ as a root. Rationale: The eigenvalue λlambdaλ of a matrix AAA is a solution to the characteristic polynomial, which is given by det(A−λI)=0text{det}(A - lambda I) = 0det(A−λI)=0. Thus, λlambdaλ 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_1mathbf{v_1} + c_2mathbf{v_2} = 0c1v1+c2v2=0 is c1=c2=0c_1 = c_2 = 0c1=c2=0. Since v1mathbf{v_1}v1 and v2mathbf{v_2}v2 point in different directions, they are indeed linearly independent. Question 7 What is the rank of the matrix B=()B = begin{pmatrix} 1 & 2 & 3 0 & 0 & 0 4 & 5 & 6 end{pmatrix}B=? A) 0 B) 1 C) 2 D) 3 Correct Answer: C) 2 Rationale: The rank of a matrix is the maximum number of linearly independent row or column vectors. Here, the first and the third rows are linearly independent, while the second row is a zero row. Thus, the rank is 2.