# Thread: Standard Matrices and basis vectors

1. ## Standard Matrices and basis vectors

I am taking an online course for linear algebra and getting a response from my teacher takes a while. Could you guys please help me understand parts a b , and d? I'm not really looking for the answer so much as the steps behind it. Thank you in advance.

2. Can you find a matrix representation of T? I think that would get you started.

3. Originally Posted by Ackbeet
Can you find a matrix representation of T? I think that would get you started.

3 -1
-1 1
0 5

I think...but what about e sub1 and e sub 2 that's what's really tripping me up.

4. I agree with your representation of $T$. Now, $e_{i}$ is a standard notation for basis vectors. I would guess that

$e_{1}=\left[\begin{matrix}1\\ 0\end{matrix}\right]$, and

$e_{2}=\left[\begin{matrix}0\\ 1\end{matrix}\right]$.

So, what do you suppose $T(e_{1})$ is?

5. The part I don't understand. Would it be a 2x2 or 3x3? I would assume it comes out to be $\begin{bmatrix}
1\, \, \, 0 & \\
0\, \, \, 1 &
\end{bmatrix}$

6. Originally Posted by Jukodan
The part I don't understand. Would it be a 2x2 or 3x3? I would assume it comes out to be $\begin{bmatrix}
1\, \, \, 0 & \\
0\, \, \, 1 &
\end{bmatrix}$
Neither. And, no, the matrix form is not $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.

You are told that the linear transfromation is "from $R^2$ to $R^3$" so you will be multiplying a matrix by a "column matrix" with 2 entries (actually 1 column, 2 rows as a matrix) so there must be two columns in you matrix to be able to multiply. The result is in $R^3$ so it will be a "column matrix" with 3 entries (1 column, 3 rows) which means your matrix must have 3 rows to give 3 results. The matrix representing this linear transformation must be "2 by 3" (2 columns, 3 rows).

Do as Ackbeet suggested: since you titled this "Standard matrices and basis vectors" he, and I, assume that $e_1= \begin{bmatrix}1 \\ 0 \end{bmatrix}$ and $e_2= \begin{bmatrix}0 \\ 1\end{bmatrix}$.

You are told that $T(x_1, y_1)= (3x_1- x_2, -x_1+ x_2, 5x_2)$ so what is T(1, 0)? What is T(0, 1)?

Now use the fact that $\begin{bmatrix}a & b\\ c & d \\ e & f\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}= \begin{bmatrix}a \\ c \\ e\end{bmatrix}$ and $\begin{bmatrix}a & b\\ c & d \\ e & f\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix}= \begin{bmatrix}b \\ d \\ f\end{bmatrix}$ to find a, b, c, d, e, and f.

7. So then $T(e_1)=\begin{bmatrix}
3\\
-1\\
0
\end{bmatrix}$
and $T(e_2)=\begin{bmatrix}
-1\\
1\\
5
\end{bmatrix}$

Am I following you correctly?

8. Yes, that is correct. And, so now what are a, b, c, d, e, f? In other words, what is the matrix representing this linear transformation in the standard basis?

9. The standard matrix is ..... $\begin{bmatrix}
3\:-1 & \\
-1\;\; 1& \\
0\;\;5&
\end{bmatrix}$

10. Originally Posted by Jukodan
The standard matrix is ..... $\begin{bmatrix}
3\:-1 & \\
-1\;\; 1& \\
0\;\;5&
\end{bmatrix}$
Yes, that is correct- very good!

11. I thank both of you very much for guiding me and shedding light on the problem. I understand this a lot better now. Both of you were very helpful, thank you again.