Determine whether the statement below is true or false. Justify the answer. Here, A is an mxn matrix. The dimension of Col A is the number of pivot columns in A. Choose the correct answer below. O A. The statement is true. The pivot columns of A form a basis for Col A. Therefore, the number of pivot columns of A is the same as the dimension of Col A. O B. The statement is false. The number of pivot columns determines the dimension of the null space, not the column space. C. The statement is false. The dimension of Col A cannot be determined without the size of matrix A. O D. The statement is true. The number of pivot columns is equal to the number of free variables in the equation Ax 0. The number of free variables in Ax = 0 is equal to the dimension of the column space.

Determine whether the statement below is true or false. Justify the answer. Here, A is an mxn matrix. The dimension of Col A is the number of pivot columns in A. Choose the correct answer below. O A. The statement is true. The pivot columns of A form a basis for Col A. Therefore, the number of pivot columns of A is the same as the dimension of Col A. O B. The statement is false. The number of pivot columns determines the dimension of the null space, not the column space. C. The statement is false. The dimension of Col A cannot be determined without the size of matrix A. O D. The statement is true. The number of pivot columns is equal to the number of free variables in the equation Ax 0. The number of free variables in Ax = 0 is equal to the dimension of the column space.

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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Question

Determine whether the statement below is true or false. Justify the answer. Here, A is an mxn matrix.
The dimension of Col A is the number of pivot columns in A.
Choose the correct answer below.
O A. The statement is true. The pivot columns of A form a basis for Col A. Therefore, the number of
pivot columns of A is the same as the dimension of Col A.
O B. The statement is false. The number of pivot columns determines the dimension of the null
space, not the column space.
C. The statement is false. The dimension of Col A cannot be determined without the size of matrix
A.
O D. The statement is true. The number of pivot columns is equal to the number of free variables in
the equation Ax = 0. The number of free variables in Ax = 0 is equal to the dimension of the
column space.

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Describe the possible echelon forms of the matrices below using 0, 1 for the pivot and * for all other entries. a. A is a 2x2 matrix with linearly independent columns. b. A is a 4x3 matrix, A= a, a, az such that {a,,a,} is linearly independent and a, is not in span{a,,a,} С. How many pivot columns must a 6x4 matrix have if its columns are independent? Why?
Determine whether the following statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 1 A matrix row operation such as -R, +R, is not permitted because of the negative fraction. Choose the correct answer below. O A. The statement is true. O B. The statement is false. A true statement is "A matrix row operation such as - R. +R, is permitted regardless of the negative fraction." 8 O C. The statement is false. A true statement is "A matrix row operation such as --R, + R, is not permitted because of the fraction 1 O D. The statement is false. A true statement is "A matrix row operation such as -R, +R, is not permitted because of the-positive fraction."
Determine if the columns of the matrix form a linearly independent set. Justify your answer. Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.) 0 -3 2 9 1 -7 4 -4 1-4-2 - 1 ⒸA. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has only the trivial solution. Therefore, the columns of A do not form a linearly independent set. B. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A do not form a linearly independent set. OC. If A is the given matrix, then the augmented atrix represents equation Ax = 0. The reduce echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A form a linearly independent set. O D.…
I believe A is correct, but I am not sure about w. It needs to be written as a fraction. Thanks!