Splitting a vector into a rowspace component and a nullspace component

In summary, the problem involves finding a basis for the orthogonal complement of the row space of matrix A. The first part of the solution involves taking A to reduced row echelon form and finding a basis for the nullspace, which is represented by the vector (-2,-2,1). However, this vector is the entire space, not a basis. For the second part, we need to find a vector in the row space and a vector in the nullspace that, when added together, equal the vector (3,3,3). To solve for this, we set up a system of equations and solve for the variables a, b, and c.
  • #1
starcoast
9
0

Homework Statement


Find a basis for the orthogonal complement of the row space of A:
A =
[1 0 2
1 1 4]

Split x = (3,3,3) into a row space component xr and a nullspace component xn.


The Attempt at a Solution


For the first part of the problem I took A to RREF
R =
[1 0 2
0 1 2]
and then solved to find a basis for the nullspace, x * (-2,-2,1), which should be the basis for the orthogonal complement of the row space.

I THINK that's right but maybe it's not because I'm stuck on the second part, splitting x = (3,3,3). Do I try to find a vector in Row(A) and a vector in Nul(A) that, when added, produce the vector (3,3,3)? How do I solve for that?

Help is very much appreciated!

Also, what's the best way to represent a matrix on these message boards?
 
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  • #2
nevermind, I figured it out
 
  • #3
an observation:

x(-2,-2,1) isn't the basis for Null(A), it's the entire space. "a" basis, is any particular vector, using a non-zero value for x. x = 1 works well, so one basis is:

{(-2,-2,1)}.

to find xn and xr, we're looking for a,b, and c with:

(3,3,3) = a(-2,-2,1) + (b(1,0,2) + c(1,1,4))

the first term is in Null(A) and the second sum of two terms in in Row(A).

this isn't that hard, we have:

(3,3,3) = (b-2a+c,c-2a,a+2b+4c), or, if you prefer:

[tex]\begin{bmatrix}-2&1&1\\-2&0&1\\1&2&4\end{bmatrix} \begin{bmatrix}a\\b\\c\end{bmatrix} = \begin{bmatrix}3\\3\\3\end{bmatrix}[/tex]

solve this for a,b and c.

****
 
Last edited by a moderator:

What does it mean to split a vector into a rowspace component and a nullspace component?

Splitting a vector into a rowspace component and a nullspace component means separating the vector into two components based on its relationship to a matrix. The rowspace component is the part of the vector that lies in the same subspace as the rows of the matrix, while the nullspace component is the part of the vector that is orthogonal to the rows of the matrix.

Why is it important to split vectors into rowspace and nullspace components?

Splitting vectors into rowspace and nullspace components allows us to better understand their relationship to a matrix. It also provides a more efficient way to perform calculations involving the vector and the matrix, as we can work with each component separately.

How do you calculate the rowspace and nullspace components of a vector?

The rowspace component of a vector can be calculated by projecting the vector onto the subspace spanned by the rows of the matrix. The nullspace component can be calculated by finding the orthogonal complement of the rowspace component, which can be done using techniques such as the Gram-Schmidt process.

What is the significance of the rowspace and nullspace components in linear algebra?

The rowspace and nullspace components play important roles in understanding the properties of a matrix and its relationship to a vector. They can also be used to solve systems of linear equations and determine the rank and nullity of a matrix.

Can you split a vector into rowspace and nullspace components for any matrix?

Yes, you can split a vector into rowspace and nullspace components for any matrix. However, the dimensions of the rowspace and nullspace components may vary depending on the properties of the matrix, such as its rank and nullity.

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