Matrix/vector equation

[TsAmE]

Let \(\displaystyle \mathbf{B} = \frac{1}{\sqrt{2}} \begin{pmatrix}1& 0\\ 1& 0\end{pmatrix}\)

Which vectors \(\displaystyle \mathbf{x}\epsilon \mathbb{R}^2\) satisfy the equation \(\displaystyle \mathbf{Bx} = \begin{pmatrix}3\\3\end{pmatrix}\)?

Attempt:

\(\displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 0\\ 1& 0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}3\\3\end{pmatrix}\)
\(\displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix}x\\x\end{pmatrix}=\begin{pmatrix}3\\3\end{pmatrix}\)

I dont know what to do now

 

[galactus]

Try \(\displaystyle 3\sqrt{2}\)

 

[TsAmE]

Try \(\displaystyle 3\sqrt{2}\)

I then got \(\displaystyle \begin{pmatrix}x\\x \end{pmatrix} = \begin{pmatrix}2\sqrt{3}\\2\sqrt{3} \end{pmatrix}\). Isnt \(\displaystyle \begin{pmatrix}x\\x \end{pmatrix}\) on the LHS wrong, as the 2nd row can only represent y i.e. \(\displaystyle \begin{pmatrix}x\\y \end{pmatrix}\)?

The correct answer was:

\(\displaystyle \boldsymbol{x} = \begin{pmatrix}3\sqrt{2}\\ t\end{pmatrix}\), t E R. I cant see where this t came from.

 

[Deleted member 4993]

Let \(\displaystyle \mathbf{B} = \frac{1}{\sqrt{2}} \begin{pmatrix}1& 0\\ 1& 0\end{pmatrix}\)

Which vectors \(\displaystyle \mathbf{x}\epsilon \mathbb{R}^2\) satisfy the equation \(\displaystyle \mathbf{Bx} = \begin{pmatrix}3\\3\end{pmatrix}\)?

Attempt:

\(\displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 0\\ 1& 0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}3\\3\end{pmatrix}\)
\(\displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix}x\\x\end{pmatrix}=\begin{pmatrix}3\\3\end{pmatrix}\)

I dont know what to do now

Now you have

\(\displaystyle \begin{pmatrix}x\\x\end{pmatrix} \ = \ \begin{pmatrix}3\sqrt{2}\\3\sqrt{2}\end{pmatrix}\)

so

x = 3?2

y = any number = t

 

[TsAmE]

x = 3?2y = any number = t

How can y = any number if you dont have a y in \(\displaystyle \begin{pmatrix}x\\x\end{pmatrix}\)?

 

[Deleted member 4993]

x = 3?2y = any number = t

How can y = any number if you dont have a y in \(\displaystyle \begin{pmatrix}x\\x\end{pmatrix}\)?

Your begining vector was {x,y} - after multiplication with matrix[A] - you had a resultant vector {x, x}, which yielded x = 3?2 - leaving y free.

Do the multiplication again with \(\displaystyle \begin{pmatrix}3\sqrt{2}\\t\end{pmatrix}\) and [A] and see what is the resultant vector.

It seems that the work you posted, was not understood by you after-all.

 

[TsAmE]

I did understand, but got confused when I saw \(\displaystyle \begin{pmatrix}x\\x \end{pmatrix}\) since I thought that the 2nd row of a matrice must be y values, 3rd row z values etc.