Change of basis

**arbolis** · August 27th 2010, 03:41 PM

In a given basis $\{ \vec e _i \}$ of a vector space, a linear transformation and a given vector of this space are determinate by $\begin{bmatrix} 2 & 1 & 0 \\ 1&2&0 \\ 0 & 0 & 5 \end{bmatrix}$ and $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ (should be a column vector).
Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by $\vec e _1 = \begin{bmatrix} 1&1&0 \end{bmatrix}$ , $\vec e _2 = \begin{bmatrix} 1&-1&0 \end{bmatrix}$ , $\vec e _3 = \begin{bmatrix} 0&0&1 \end{bmatrix}$ (they should be column vectors).

My attempt: I formed a matrix whose columns are the $\vec e_1$ , $\vec e_2$ , $\vec e_3$ and I found its inverse. Call them $S$ and $S^{-1}$ .
Then I multiplied $S^{-1}AS$ where $A$ is the first matrix I wrote. As result, I obtained the following matrix $A'= \begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&5 \end{bmatrix}$ which would be the matrix they asked for.

For the vector, I wasn't sure at all... I just multipled $A'$ by $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ and I obtained $\begin{bmatrix} 3 & 12 & 15 \end{bmatrix}$ which would be the vector they ask for.
Am I right?

**HallsofIvy** · August 29th 2010, 03:08 PM

Originally Posted by arbolis

In a given basis $\{ \vec e _i \}$ of a vector space, a linear transformation and a given vector of this space are determinate by $\begin{bmatrix} 2 & 1 & 0 \\ 1&2&0 \\ 0 & 0 & 5 \end{bmatrix}$ and $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ (should be a column vector).
Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by $\vec e _1 = \begin{bmatrix} 1&1&0 \end{bmatrix}$ , $\vec e _2 = \begin{bmatrix} 1&-1&0 \end{bmatrix}$ , $\vec e _3 = \begin{bmatrix} 0&0&1 \end{bmatrix}$ (they should be column vectors).

My attempt: I formed a matrix whose columns are the $\vec e_1$ , $\vec e_2$ , $\vec e_3$ and I found its inverse. Call them $S$ and $S^{-1}$ .
Then I multiplied $S^{-1}AS$ where $A$ is the first matrix I wrote. As result, I obtained the following matrix $A'= \begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&5 \end{bmatrix}$ which would be the matrix they asked for.

For the vector, I wasn't sure at all... I just multipled $A'$ by $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ and I obtained $\begin{bmatrix} 3 & 12 & 15 \end{bmatrix}$ which would be the vector they ask for.
Am I right?

The first part, A', is correct but the question did not ask for the vector <1 2 3> multiplied by A'.

Suppose your problem were Av= u where [tex]A= \begin{barray}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 5\end{bmatrix} and [tex]v= \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}.

To change A to A' you would, as you say, write $S^{-1}AS$ with the S matrix you give. Now think how we would apply that to the equation Av= u. If we multiply both sides, on the left, by $S^{-1}$ , we would have $S^{-1}Av= S^{-1}u$ . But that does not have the "S" multiplying A on the right. Just multiplying the equation on the right won't help because that would give $S^{-1}AvS= S^{-1}uS$ , not what we want.

Instead, use the fact that $I= SS^{-1}$ to write the equation as $S^{-1}A(SS^{-1})v= S^{-1}u$ or $(S^{-1}AS)(S^{-1}v)= S^{-1}u$ - that is, $A'v'= u'$ with $A'= S^{-1}AS$ and $v'= S^{-1}v$ , not A'v.

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