Linear Algebra proof, diagonalization

[I like Serena]

Suppose $N=\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix}$.
Does it satisfy the problem criteria?
Are V_1 and V_2 as you're suggesting?

 

[hedgie]

Do they satisfy the transformation or the ((1,1),(-1,-1)) * V_1= V_2...no for the second one if V_1 = (0,1)...completely lost now.

 

[I like Serena]

Suppose V_1=(1,1).
Does it fit?

What would V_2 be?
And what would N^2 be?

 

[hedgie]

Are you just picking arbitrary V_1? shouldn't it be the first column?

 

[I like Serena]

No, it's not arbitrary. :)

Why should it be the first column?

 

[hedgie]

We had always used columns in class and from the book...and the basis are all columns and everything we have been writing or I at least have been writing is in columns.

Which is why I would've thought (1,-1) is V1 and why I asked if it was arbitrary...if I was using rows i'd expect (1,1).

 

[I like Serena]

That's why I adjusted post #36, to avoid confusion.

So what is:
$$N\cdot \begin{pmatrix}1 \\ 1\end{pmatrix}=\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix} \cdot \begin{pmatrix}1 \\ 1\end{pmatrix}$$

And what do you get if you multiply N with the resulting vector?

What are therefore V_1 and V_2?

 

[hedgie]

(a_11+a_12,a_21+a_22) as a 2x1 matrix if we do not know what v_1 and v_2 are..

V_1 and V_2 are both (1,1) or the 2x2 matrix of ((1,1),(1,1))

 

[I like Serena]

Did you calculate $\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix} \cdot \begin{pmatrix}1 \\ 1\end{pmatrix}$?

 

[hedgie]

yes (2,-2) column vector...also tried to solve the problem using the matrix ((a,b),(c,d)) and got a=+-d

 

[I like Serena]

yes (2,-2) column vector

Yes. And N*(2,-2)?

...also tried to solve the problem using the matrix ((a,b),(c,d)) and got a=+-d

a=-d would be right, but how did you get a=d?

 

[hedgie]

I don't, I only got a=-d, sorry typo.

((1,-1),(1,-1))*(2,-2)=(0,0) and (V_1 V_2) = ((2,-2),(2,-2))

 

[I like Serena]

I don't, I only got a=-d, sorry typo.

Yes.

((1,-1),(1,-1))*(2,-2)=(0,0) and (V_1 V_2) = ((2,-2),(2,-2))

No: (V_1 V_2) = ((1,1),(2,-2))

 

[hedgie]

Why would it be the first row and the column that i am multiplying it against transposed to a row?