6.1: Linear Transformations
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Outcomes
- Understand the definition of a linear transformation, and that all linear transformations are determined by matrix multiplication.
Recall that when we multiply an \(m\times n\) matrix by an \(n\times 1\) column vector, the result is an \(m\times 1\) column vector. In this section we will discuss how, through matrix multiplication, an \(m \times n\) matrix transforms an \(n\times 1\) column vector into an \(m \times 1\) column vector.
Recall that the \(n \times 1\) vector given by \[\vec{x} = \left [ \begin{array}{r} x_1 \\ x_2\\ \vdots \\ x_n \end{array} \right ]\nonumber \] is said to belong to \(\mathbb{R}^n\), which is the set of all \(n \times 1\) vectors. In this section, we will discuss transformations of vectors in \(\mathbb{R}^n.\)
Consider the following example.
Example \(\PageIndex{1}\): A Function Which Transforms Vectors
Consider the matrix \(A = \left [ \begin{array}{ccc} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] .\) Show that by matrix multiplication \(A\) transforms vectors in \(\mathbb{R}^3\) into vectors in \(\mathbb{R}^2\).
Solution
First, recall that vectors in \(\mathbb{R}^3\) are vectors of size \(3 \times 1\), while vectors in \(\mathbb{R}^{2}\) are of size \(2 \times 1\). If we multiply \(A\), which is a \(2 \times 3\) matrix, by a \(3 \times 1\) vector, the result will be a \(2 \times 1\) vector. This what we mean when we say that \(A\) transforms vectors.
Now, for \(\left [ \begin{array}{c} x \\ y \\ z \end{array} \right ]\) in \(\mathbb{R}^3\), multiply on the left by the given matrix to obtain the new vector. This product looks like \[\left [ \begin{array}{rrr} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{r} x \\ y \\ z \end{array} \right ] = \left [ \begin{array}{c} x+2y \\ 2x+y \end{array} \right ]\nonumber \] The resulting product is a \(2 \times 1\) vector which is determined by the choice of \(x\) and \(y\). Here are some numerical examples. \[\left [ \begin{array}{ccc} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right ] = \ \left [ \begin{array}{c} 5 \\ 4 \end{array} \right ]\nonumber \] Here, the vector \(\left [ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right ]\) in \(\mathbb{R}^3\) was transformed by the matrix into the vector \(\left [ \begin{array}{c} 5 \\ 4 \end{array}\right ]\) in \(\mathbb{R}^2\).
Here is another example: \[\left [ \begin{array}{rrr} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{r} 10 \\ 5 \\ -3 \end{array} \right ] = \ \left [ \begin{array}{r} 20 \\ 25 \end{array} \right ]\nonumber \]
The idea is to define a function which takes vectors in \(\mathbb{R}^{3}\) and delivers new vectors in \(\mathbb{R}^{2}.\) In this case, that function is multiplication by the matrix \(A\).
Let \(T\) denote such a function. The notation \(T:\mathbb{R}^{n}\mapsto \mathbb{R}^{m}\) means that the function \(T\) transforms vectors in \(\mathbb{R}^{n}\) into vectors in \(\mathbb{R}^{m}\). The notation \(T(\vec{x})\) means the transformation \(T\) applied to the vector \(\vec{x}\). The above example demonstrated a transformation achieved by matrix multiplication. In this case, we often write \[T_{A}\left( \vec{x}\right) =A \vec{x}\nonumber \] Therefore, \(T_{A}\) is the transformation determined by the matrix \(A\). In this case we say that \(T\) is a matrix transformation.
Recall the property of matrix multiplication that states that for \(k\) and \(p\) scalars, \[A\left( kB+pC\right) =kAB+pAC\nonumber \] In particular, for \(A\) an \(m\times n\) matrix and \(B\) and \(C,\) \(n\times 1\) vectors in \(\mathbb{R}^{n}\), this formula holds.
In other words, this means that matrix multiplication gives an example of a linear transformation, which we will now define.
Definition \(\PageIndex{1}\): Linear Transformation
Let \(T:\mathbb{R}^{n}\mapsto \mathbb{R}^{m}\) be a function, where for each \(\vec{x} \in \mathbb{R}^{n},T\left(\vec{x}\right)\in \mathbb{R}^{m}.\) Then \(T\) is a linear transformation if whenever \(k ,p\) are scalars and \(\vec{x}_1\) and \(\vec{x}_2\) are vectors in \(\mathbb{R}^{n}\) \(( n\times 1\) vectors\(),\) \[T\left( k \vec{x}_1 + p \vec{x}_2 \right) = kT\left(\vec{x}_1\right)+ pT\left(\vec{x}_{2} \right)\nonumber \]
Consider the following example.
Example \(\PageIndex{2}\): Linear Transformation
Let \(T\) be a transformation defined by \(T:\mathbb{R}^3\to\mathbb{R}^2\) is defined by \[T\left [\begin{array}{c} x \\ y \\ z \end{array}\right ] = \left [\begin{array}{c} x+y \\ x-z \end{array}\right ] \mbox{ for all } \left [\begin{array}{c} x \\ y \\ z \end{array}\right ] \in\mathbb{R}^3\nonumber \] Show that \(T\) is a linear transformation.
Solution
By Definition \(\PageIndex{1}\) we need to show that \(T\left( k \vec{x}_1 + p \vec{x}_2 \right) = kT\left(\vec{x}_1\right)+ pT\left(\vec{x}_{2} \right)\) for all scalars \(k,p\) and vectors \(\vec{x}_1, \vec{x}_2\). Let \[\vec{x}_1 = \left [\begin{array}{c} x_1 \\ y_1 \\ z_1 \end{array}\right ], \vec{x}_2 = \left [\begin{array}{c} x_2 \\ y_2 \\ z_2 \end{array}\right ]\nonumber \] Then \[\begin{aligned} T\left( k \vec{x}_1 + p \vec{x}_2 \right) &= T \left( k \left [\begin{array}{c} x_1 \\ y_1 \\ z_1 \end{array}\right ] + p \left [\begin{array}{c} x_2 \\ y_2 \\ z_2 \end{array}\right ] \right) \\ &= T \left( \left [\begin{array}{c} kx_1 \\ ky_1 \\ kz_1 \end{array}\right ] + \left [\begin{array}{c} px_2 \\ py_2 \\ pz_2 \end{array}\right ] \right) \\ &= T \left( \left [\begin{array}{c} kx_1 + px_2 \\ ky_1 + py_2 \\ kz_1 + pz_2 \end{array}\right ] \right) \\ &= \left [\begin{array}{c} (kx_1 + px_2) + (ky_1 + py_2) \\ (kx_1 + px_2)- (kz_1 + pz_2) \end{array}\right ] \\ &= \left [\begin{array}{c} (kx_1 + ky_1) + (px_2 + py_2) \\ (kx_1 - kz_1) + (px_2 - pz_2) \end{array}\right ] \\ &= \left [\begin{array}{c} kx_1 + ky_1 \\ kx_1 - kz_1 \end{array}\right ] + \left [ \begin{array}{c} px_2 + py_2 \\ px_2 - pz_2 \end{array}\right ] \\ &= k \left [\begin{array}{c} x_1 + y_1 \\ x_1 - z_1 \end{array}\right ] + p \left [ \begin{array}{c} x_2 + y_2 \\ x_2 - z_2 \end{array}\right ] \\ &= k T(\vec{x}_1) + p T(\vec{x}_2) \end{aligned}\nonumber \] Therefore \(T\) is a linear transformation.
Two important examples of linear transformations are the zero transformation and identity transformation. The zero transformation defined by \(T\left( \vec{x} \right) = \vec{0}\) for all \(\vec{x}\) is an example of a linear transformation. Similarly the identity transformation defined by \(T\left( \vec{x} \right) = \vec{x}\) is also linear. Take the time to prove these using the method demonstrated in Example \(\PageIndex{2}\) .
We began this section by discussing matrix transformations, where multiplication by a matrix transforms vectors. These matrix transformations are in fact linear transformations.
It turns out that every linear transformation can be expressed as a matrix transformation, and thus linear transformations are exactly the same as matrix transformations.