Vectors and spans physics help

[hkus10]

Determine whether the given vector A in 2x2 matrix belongs to span{A1, A2, A3}, where
A1 =
[1 -1
0 3]

A2 =
[1 1
0 2]

A3 =
[2 2
-1 1]

A =
[5 1
-1 9].

Since A1, A2, A3 are not a nx1 matrices, I cannot put this into reduced echelon form? Therefore, what can I do to solve this problem?

Thanks

 

[Kreizhn]

You can view this as a problem in $\mathbb R^4$, or simply as a set of vectors in $M_{2\times 2}(\mathbb R)$.

Namely, if you want to use reduced row echelon form, note that you can write

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b & c & d \end{pmatrix}$$

You are allowed to do addition and subtraction here like normal which will allow you to determine the span. However, be warned that normal multiplication cannot be done, so be careful.

 

[hkus10]

You can view this as a problem in $\mathbb R^4$, or simply as a set of vectors in $M_{2\times 2}(\mathbb R)$.

Namely, if you want to use reduced row echelon form, note that you can write

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b & c & d \end{pmatrix}$$

You are allowed to do addition and subtraction here like normal which will allow you to determine the span. However, be warned that normal multiplication cannot be done, so be careful.

Just want to clarify the way it works:
With the scalar that I have in each vector,
Can I put into this form in the following?
[
1 1 2 5
-1 1 2 1
0 0 -1 -1
3 2 1 9]

or
[
1 -1 0 3
1 1 0 2
2 2 -1 1
5 1 -1 9]

or what is it?

 

[Kreizhn]

Sorry, I shouldn't have written it as a row, it should be a column for you to apply REF. What you are doing is identifying each matrix as a "vector," so to apply to REF you want to make each matrix into a column vector.

However, if you know what the row-space and column space are, you make the matrix into either a column or a row.

Furthermore, note that you can do

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a \\ c \\ b \\ d \end{pmatrix}$$
OR
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \\ d\end{pmatrix}$$
Or really any other combination you like, just so long as you are consistent when you convert back into its matrix form.

 

[Kreizhn]

Give it a try and see what works. There is an important conceptual step going on here, namely that the vector space $(M_{n\times m}(\mathbb F),+)$ of $n \times m$ matrices over the field $\mathbb F$ under point-wise matrix addition, is isomorphic to the vector space $(\mathbb F^{nm}, +)$ under normal vector addition.