Question

Find the basis for N(A) and R(A) of the matrix A = [ 4 8 1 1 6 3 5 1 2 5 2 4 1 9 10 1 2 3 2 0 ] To find the basis for N(A), we need to find the null space of A. This can be done by solving the equation Ax = 0, where x is a vector. To find the basis for R(A), we need to find the column space of A. This can be done by finding the pivot columns of A. Let's start by finding the null space of A: To solve the equation Ax = 0, we can row reduce the augmented matrix [A | 0]: [ 4 8 1 | 0 1 6 3 | 0 5 1 2 | 0 5 2 4 | 0 1 9 10 | 0 1 2 3 | 0 2 0 | 0 ] Using row operations, we can simplify the matrix: [ 1 2 3 | 0 0 1 -1 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 | 0 ] From this row reduced form, we can see that the third and fourth columns are free variables. Therefore, the basis for N(A) is the vectors: [ -2 1 1 ] [ -3 1 0 ] Now let's find the column space of A: To find the pivot columns of A, we can row reduce the matrix A: [ 4 8 1 1 6 3 5 1 2 5 2 4 1 9 10 1 2 3 2 0 ] Using row operations, we can simplify the matrix: [ 1 2 3 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] From this row reduced form, we can see that the first and second columns are pivot columns. Therefore, the basis for R(A) is the vectors: [ 1 0 0 0 0 0 0 ] [ 2 1 0 0 0 0 0 ]

Name: find the basis for na and ra of the matrix a 4 8 1 1 6 3 5 1 2 5 2 4 1 9 10 1 2 3 2 0 to find the basis for na we need to find the null space of a this can be done by solving the equation ax 47766
Uploaded: 2023-09-15T12:15:29-08:00
Duration: 4 min 55 s
Channel: Adi S
Description: find the basis for na and ra of the matrix a 4 8 1 1 6 3 5 1 2 5 2 4 1 9 10 1 2 3 2 0 to find the basis for na we need to find the null space of a this can be done by solving the equation ax 47766

          Find the basis for N(A) and R(A) of the matrix A = 
[ 4 8 1 
  1 6 3 
  5 1 2 
  5 2 4 
  1 9 10 
  1 2 3 
  2 0 ]

To find the basis for N(A), we need to find the null space of A. This can be done by solving the equation Ax = 0, where x is a vector.

To find the basis for R(A), we need to find the column space of A. This can be done by finding the pivot columns of A.

Let's start by finding the null space of A:

To solve the equation Ax = 0, we can row reduce the augmented matrix [A | 0]:

[ 4 8 1 | 0 
  1 6 3 | 0 
  5 1 2 | 0 
  5 2 4 | 0 
  1 9 10 | 0 
  1 2 3 | 0 
  2 0 | 0 ]

Using row operations, we can simplify the matrix:

[ 1 2 3 | 0 
  0 1 -1 | 0 
  0 0 0 | 0 
  0 0 0 | 0 
  0 0 0 | 0 
  0 0 0 | 0 
  0 0 | 0 ]

From this row reduced form, we can see that the third and fourth columns are free variables. Therefore, the basis for N(A) is the vectors:

[ -2 
   1 
   1 ]

[ -3 
   1 
   0 ]

Now let's find the column space of A:

To find the pivot columns of A, we can row reduce the matrix A:

[ 4 8 1 
  1 6 3 
  5 1 2 
  5 2 4 
  1 9 10 
  1 2 3 
  2 0 ]

Using row operations, we can simplify the matrix:

[ 1 2 3 
  0 1 -1 
  0 0 0 
  0 0 0 
  0 0 0 
  0 0 0 
  0 0 0 ]

From this row reduced form, we can see that the first and second columns are pivot columns. Therefore, the basis for R(A) is the vectors:

[ 1 
  0 
  0 
  0 
  0 
  0 
  0 ]

[ 2 
  1 
  0 
  0 
  0 
  0 
  0 ]

Added by George E.

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