QUICK QUESTION about matrix (linear algebra)

[kougou]

Hello guys,
I am wondering, could a elementary row operation be done to a matrix without changing it.

I know that if we have the SYSTEM, then we could exchange rows, scalar multiplication of a row, linear combination of one rows.

but if we are only given a matrix, and we are asked to row reduce that matrix, could we still do scalar multiplication of a rows, exchange rows, etc?

Thank you

 

[vela]

Yup. You can think of row-reducing a matrix by itself as essentially solving the system Ax=0, which is something you might want to do to show that vectors are linearly independent, that the kernel of a linear map is trivial, etc. There's really no point in tacking on a column of zeros to form an enhanced matrix since the row operations will leave it unchanged.

 

[kougou]

Yup. You can think of row-reducing a matrix by itself as essentially solving the system Ax=0, which is something you might want to do to show that vectors are linearly independent, that the kernel of a linear map is trivial, etc. There's really no point in tacking on a column of zeros to form an enhanced matrix since the row operations will leave it unchanged.

Thank you for your reply.
I remember my TA told me that when we are doing row reduction for a matrix, we must use this symbol "→", not "=", or else marks will be reduced.
I ask him why, but I not exactly remember what he said.

so, I got confuse, ... because of this symbol, sometimes when I am asked to reduce a matrix, I afraid of exchanging rows, etc, because I thought it's not a system.

thank you, very helpful

 

[vela]

Your TA said that because the equal sign has a specific meaning. If you write, for example,
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},$$ it means that a=1, b=2, c=3, and d=4. If you have the matrix $\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$ and you subtract row 1 from row 2, it wouldn't make sense to write
$$\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$$ because you'd be saying that 1=0 and 3=1, which is obviously nonsense.

 

[Mark44]

The symbol for equivalence is used more often, I believe. In fact, I don't recall seeing an arrow used in this context.

Using vela's example, we would have
$$\begin{bmatrix}1&2 \\ 1 & 3\end{bmatrix} \equiv \begin{bmatrix}1&2 \\ 0 & 1\end{bmatrix}$$

Here the two matrices are equivalent.
The first matrix could represent the system of equations
x + 2y = 0
x + 3y = 0

The system above is equivalent to this system:
x + 2y = 0
0x + y = 0

Notice that it wouldn't make any sense to call the two systems of equations "equal."