13 is a linear transformation and .......Determine T

In summary, we are given a linear transformation $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ and three vectors $v_1, v_2, v_3$ such that $T(v_1) = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}$, $T(v_2) = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}$, and $T(v_3) = \begin{bmatrix} 2 \\2 \\3 \\ \end{bmatrix}$. We are asked to determine $T\begin{bmatrix} 1 \\2
  • #1
karush
Gold Member
MHB
3,269
5
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and
$$T \begin{bmatrix}
1 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\2 \\1 \\
\end{bmatrix},
\quad T \begin{bmatrix}
1 \\0 \\1 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\0 \\2 \\
\end{bmatrix}, \quad
T \begin{bmatrix}
0 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
2 \\2 \\3 \\
\end{bmatrix}.$$
Determine $T
\begin{bmatrix}
1 \\2 \\3 \\
\end{bmatrix}$

ok this should be easy... but.. the examples were not that close to this
I presume we could start with the middle one.
 
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  • #2
karush said:
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and
$$T \begin{bmatrix}
1 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\2 \\1 \\
\end{bmatrix},
\quad T \begin{bmatrix}
1 \\0 \\1 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\0 \\2 \\
\end{bmatrix}, \quad
T \begin{bmatrix}
0 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
2 \\2 \\3 \\
\end{bmatrix}.$$
Determine $T
\begin{bmatrix}
1 \\2 \\3 \\
\end{bmatrix}$

ok this should be easy... but.. the examples were not that close to this
I presume we could start with the middle one.
Well, the most direct method would be to simply solve for T. But since T is linear there is another way.

Can you build \(\displaystyle \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] \) out of a linear combination of \(\displaystyle \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] \), \(\displaystyle \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] \), and \(\displaystyle \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ] \)?

-Dan
 
  • #3
ok this probably is not exactly what it is supposed to be but

$T \begin{bmatrix} 1 \\1 \\0 \end{bmatrix}
=\begin{bmatrix} 1 \\2 \\1 \end{bmatrix},
\quad T \begin{bmatrix} 1 \\0 \\1 \end{bmatrix}
= \begin{bmatrix} 1 \\0 \\2 \end{bmatrix}'
\quad T \begin{bmatrix} 0 \\1 \\0 \end{bmatrix}
= \begin{bmatrix} 2 \\2 \\3 \end{bmatrix}$.

$=T\left[\begin{array}{c}
1&1&0\\
1&0&1\\
0&1&0
\end{array}\right]$

or possibly
$\left[\begin{array}{c}x_1+x_2 \\x_1+x_3\\x_2\end{array}\right]$
 
Last edited:
  • #4
karush said:
ok this probably is not exactly what it is supposed to be but

$T \begin{bmatrix} 1 \\1 \\0 \end{bmatrix}
=\begin{bmatrix} 1 \\2 \\1 \end{bmatrix},
\quad T \begin{bmatrix} 1 \\0 \\1 \end{bmatrix}
= \begin{bmatrix} 1 \\0 \\2 \end{bmatrix}'
\quad T \begin{bmatrix} 0 \\1 \\0 \end{bmatrix}
= \begin{bmatrix} 2 \\2 \\3 \end{bmatrix}$.

$=T\left[\begin{array}{c}
1&1&0\\
1&0&1\\
0&1&0
\end{array}\right]$

are we trying to build $Ax=B$
What I am trying to point you toward is
\(\displaystyle \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ] \)

so
\(\displaystyle T \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a T \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b T \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c T \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ] \)

Note that this can only be done if T is linear.

-Dan
 
  • #5
then?
$\left[\begin{array}{c}Ta+Tb \\Ta+Tc\\Tb\end{array}\right]$i think I am getting confused by looking at too many examples

how would we know if T is linear?
 
  • #6
karush said:
how would we know if T is linear?
karush said:
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation

karush said:
then?
topsquark said:
\(\displaystyle T \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a T \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b T \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c T \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ] \)
and
karush said:
$$T \begin{bmatrix}
1 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\2 \\1 \\
\end{bmatrix},
\quad T \begin{bmatrix}
1 \\0 \\1 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\0 \\2 \\
\end{bmatrix}, \quad
T \begin{bmatrix}
0 \\1 \\0 \\
\end{bmatrix}
=
\begin{bmatrix}
2 \\2 \\3 \\
\end{bmatrix}.$$

As for
karush said:
$\left[\begin{array}{c}Ta+Tb \\Ta+Tc\\Tb\end{array}\right]$
$Ta$ does not makes sense for $a\in\mathbb{R}$ because $T:\mathbb{R}^3\to\mathbb{R}^3$.
 
  • #7
ok apparently I'm not understanding the steps
not sure what I should be asking
 

Related to 13 is a linear transformation and .......Determine T

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the linear structure of the original space. This means that the transformation follows certain rules, such as preserving addition and scalar multiplication.

2. How can I determine if T is a linear transformation?

To determine if T is a linear transformation, you need to check if it follows the two main properties of linearity: additivity and homogeneity. Additivity means that T(u + v) = T(u) + T(v), and homogeneity means that T(cu) = cT(u), where u and v are vectors and c is a scalar. If T satisfies both of these properties, it is a linear transformation.

3. What is the importance of linear transformations?

Linear transformations are important in mathematics and science because they provide a way to model and understand real-world phenomena. They are also used in various fields such as computer graphics, economics, and physics to solve problems and make predictions.

4. Can a linear transformation be represented by a matrix?

Yes, a linear transformation can be represented by a matrix. This is known as the standard matrix representation of a linear transformation. The columns of the matrix represent the images of the standard basis vectors, and the matrix can be used to perform calculations involving the transformation.

5. How can I determine the dimension of the output space of T?

The dimension of the output space of T can be determined by finding the number of linearly independent columns in the standard matrix representation of T. This is known as the rank of the matrix and is equal to the dimension of the output space. Alternatively, you can also find the dimension of the output space by taking the number of rows in the matrix and subtracting the nullity, which is the dimension of the null space or kernel of T.

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