A [b]transformation[/b] is simply a [b]function[/b] (or [b]mapping[/b]) [math]T[/math] from [math]\mathbb{R}^n[/math] to [math]\mathbb{R}^m[/math] i.e. for any (input) vector [math]v[/math] in [math]\mathbb{R}^n[/math], [math]T(v)[/math] is an (output) vector in [math]\mathbb{R}^m[/math]. In linear algebra, we will mainly study a very special type of transformations called [b]linear transformations[/b]. They are transformations that [u]preserve vector addition and scaling[/u] i.e. [math]T:\mathbb{R}^n\to\mathbb{R}^m[/math] is a linear transformation if for any vectors [math]u, v[/math] in [math]\mathbb{R}^n[/math] and any real number [math]k[/math], we have [br][br][math]T\left(u+v\right)=T\left(u\right)+T\left(v\right)[/math], and [math]T\left(kv\right)=kT\left(v\right)[/math][br][br]Therefore, given the standard basis [math]e_1,e_2, \ldots, e_n[/math] for [math]\mathbb{R}^n[/math], for any vector [math]v[/math] in [math]\mathbb{R}^n[/math], it can be written as a linear combination of the standard basis as follows:[br][br][math]v=c_1e_1+c_2e_2+\cdots+c_ne_n[/math][br][br]When we apply a linear transformation [math]T[/math] to it, using the fact that it preserves vector addition and scaling, we get[br][br][math]T(v)=c_1T(e_1)+c_2T(e_2)+\cdots+c_nT(e_n)[/math][br][br]You can see that [math]T(v)[/math] is the linear combination of [math]T(e_1),T(e_2),\ldots,T(e_n)[/math] with the same weight. In other words, the linear transformation [math]T[/math] is uniquely determined by [math]T(e_1),T(e_2),\ldots,T(e_n)[/math].[br][br]In the following applet, we consider any linear transformation [math]T:\mathbb{R}^2\to\mathbb{R}^2[/math]. As mentioned above, we can define [math]T[/math] by specifying [math]T(\hat{\mathbf{i}})[/math] and [math]T(\hat{\mathbf{j}})[/math]. You can change [math]T(\hat{\mathbf{i}})[/math] and [math]T(\hat{\mathbf{j}})[/math] freely, then click the "Go" button and see how the grid in the domain is "transformed" under [math]T[/math]. Also, you can define the vector [math]v[/math] by inputting the coordinates of its arrowhead. The column vector [math]T(v)[/math] will be shown. Again, you can click the "Go" button to see the transformation from [math]v[/math] to [math]T(v)[/math] visually.[br][br]
The following are some questions that test your understanding on linear transformations:
Will a grid always be transformed into a grid by any linear transformation? Explain your answer briefly.
No. If u and v are linearly dependent, then a grid will not be transformed into a grid. For example, if u and v are in the same (or opposite direction), the grid will be transformed to points along the line containing u and v. If u and v are both zero vectors, the grid will be transformed to the single point at the origin![br][br]However, if we further assume that u and v are linearly independent, then a grid will be transformed into then new grid generated by [math]T(\hat{\mathbf{i}})[/math] and [math]T(\hat{\mathbf{j}})[/math]. It is because any grid point [math]n\hat{\mathbf{i}}+m\hat{\mathbf{j}}[/math], where [math]m, n[/math] are integers, is transformed to [math]nT(\hat{\mathbf{i}})+mT(\hat{\mathbf{j}})[/math], which is the corresponding point of the new grid.
Can a zero vector be transformed to a non-zero vector by a linear transformation? Explain your answer briefly.
No. For any linear transformation [math]T:\mathbb{R}^n\to\mathbb{R}^m[/math], we have[br][br][math]T(0)=T(0e_1+0e_2+\cdots+0e_n)=0T(e_1)+0T(e_2)+\cdots+0T(e_n)=0[/math][br][br]Hence, any linear transformation transforms zero vector to zero vector.
Which of the following is a linear transformation from [math]\mathbb{R}^2[/math] to [math]\mathbb{R}^2[/math]? You can select more than one answer.
Let [math]T:\mathbb{R}^2\to\mathbb{R}^2[/math] such that [math]T(\hat{\mathbf{i}})=\begin{pmatrix}3 \\ -1\end{pmatrix}[/math] and [math]T(\hat{\mathbf{j}})=\begin{pmatrix}2 \\ 0\end{pmatrix}[/math]. Find [math]T\left(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\right)[/math].[br](You should first try to compute for the answer without using the above applet.)
[math]\begin{eqnarray}T\left(\begin{pmatrix}-2\\3\end{pmatrix}\right)&=&T(-2\hat{\mathbf{i}}+3\hat{\mathbf{j}}) \\ &=& -2T(\hat{\mathbf{i}})+3T(\hat{\mathbf{j}})\\[br]&=&-2\begin{pmatrix}3\\-1\end{pmatrix}+3\begin{pmatrix}2\\0\end{pmatrix}\\[br]&=&\begin{pmatrix}0\\2\end{pmatrix}\end{eqnarray}[/math]
Let [math]T:\mathbb{R}^3\to\mathbb{R}^2[/math] such that [math]T(\hat{\mathbf{i}})=\begin{pmatrix}2 \\ -3\end{pmatrix}[/math], [math]T(\hat{\mathbf{j}})=\begin{pmatrix}1 \\ -4\end{pmatrix}[/math] and [math]T(\hat{\mathbf{k}})=\begin{pmatrix}0 \\ 5\end{pmatrix}[/math]. Find [math]T\left(\begin{pmatrix} 3 \\ -1 \\ 2\end{pmatrix}\right)[/math].[br]
[math]\begin{eqnarray}T\left(\begin{pmatrix}3\\-1\\2\end{pmatrix}\right)&=&T(3\hat{\mathbf{i}}-\hat{\mathbf{j}}+2\hat{\mathbf{k}}) \\ &=& 3T(\hat{\mathbf{i}})-T(\hat{\mathbf{j}})+2T(\hat{\mathbf{k}})\\[br]&=&3\begin{pmatrix}2\\-3\end{pmatrix}-\begin{pmatrix}1\\-4\end{pmatrix}+2\begin{pmatrix}0\\5\end{pmatrix}\\[br]&=&\begin{pmatrix}5\\5\end{pmatrix}\end{eqnarray}[/math]