How Can You Combine Bases from Subspaces in Linear Algebra?

[applechu]

Hi:
I have a problem about combine bases from subspaces. This is part of orthogonality.
The examples as following:
For A=$\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$ split x= $\begin{bmatrix} 4 \\ 3 \end{bmatrix}$ into $x_r$+$x_n$=$\begin{bmatrix} 2 \\ 4 \end{bmatrix}+\begin{bmatrix} 2 \\ -1 \end{bmatrix}$

I don't know why it can split into $x_r$+$x_n$, and how to prove that,
thanks a lot

 

[algebrat]

It looks like you haven't given us everything from your notes or problem, but I'll try to infer what is meant.

(2,-1) is a basis vector for the null space.

Orthogonal to that space is the space with spanning set (1,2)

(the slope is the negative reciprocal)

Then any vector, such as (4,3), can be written as a linear combination of these.

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

Then solving the system of two linear equations for the two unknowns a and b, we get a=1, b=2.

Or was there something else you were trying to "prove"?

 

[applechu]

In fact, I feel I have stuck into the situation of learning linear algebra.
I read the part of orthogonality and four subspaces.
I feel confused about some examples, such as following:
B=$\begin{bmatrix} 1 & 2&3&4&5 \\ 1 & 2&4&5&6 \\ 1 & 2&4&5&6 \end{bmatrix}$ conatins
$\begin{bmatrix} 1 &3 \\ 1& 4\end{bmatrix}$ in the pivot rows and columns.
However, I can not deduce the process how it from. I try elimination, but it is not so directly perceived through the notation of submatrix, thanks

 

[algebrat]

In fact, I feel I have stuck into the situation of learning linear algebra.
I read the part of orthogonality and four subspaces.
I feel confused about some examples, such as following:
B=$\begin{bmatrix} 1 & 2&3&4&5 \\ 1 & 2&4&5&6 \\ 1 & 2&4&5&6 \end{bmatrix}$ conatins
$\begin{bmatrix} 1 &3 \\ 1& 4\end{bmatrix}$ in the pivot rows and columns.
However, I can not deduce the process how it from. I try elimination, but it is not so directly perceived through the notation of submatrix, thanks

The form

$\begin{bmatrix} 1 & 2&3&4&5 \\ 0 & 0&1&1&1 \\ 0&0&0&0&0 \end{bmatrix}$, or $\begin{bmatrix} 1 & 2&0&1&2 \\ 0 & 0&1&1&1 \\ 0&0&0&0&0 \end{bmatrix}$

Tell you to take the first and third columns as a basis for the column space,

$\begin{bmatrix} 1 \\ 1\\1\end{bmatrix}$ and $\begin{bmatrix} 3 \\ 4\\4\end{bmatrix}$

The last look similar to the 2 by 2 matrix you wrote, but I don't understand why you would want that matrix, can you tell us more about the problem?

 

[applechu]

It is a example from the book. I try to learn linear algebra from some books.
thanks a lot.

 

[HallsofIvy]

You say "I feel I have stuck into the situation of learning linear algebra." What you give, "subspaces", "basis", etc. is linear algebra. What course was this for?

 

[applechu]

For the chapter about orthogonality. Thanks.

 

[HallsofIvy]

I asked what course, if not Linear Algebra, not what chapter.