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Topos Uranos
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Matrices as transformations of the plane

Summary

This lesson interprets each matrix with two rows and two columns as a transformation of the plane, the one that takes each v⃗\vec{v}, written as a column, to the vector Av⃗A\vec{v}. It proves that these transformations are exactly the linear ones and that each one is determined by the images of the two standard basis vectors, which are the columns of its matrix. With that principle it obtains the matrices of the rotations, of the projections onto lines through the origin and of the reflections in them, of the dilations and of the shears. It proves that composing transformations amounts to multiplying their matrices, from which follow the addition of the angles of two rotations, the fact that two composed reflections form a rotation through twice the angle between their lines, and the fact that reflections do not commute in general. It interprets the determinant as the factor by which the transformation multiplies areas, with a negative sign when it reverses the orientation, and characterizes the bijective transformations as those with nonzero determinant. It ends by rewriting in matrix form the rotation of axes of the general second-degree equation.

Learning objectives

  1. Recognize a matrix as a linear transformation of the plane and obtain the matrix of a transformation from the images of the standard basis vectors.
  2. Write and derive the matrices of rotations, projections, reflections, dilations and shears, and describe geometrically the effect of a given matrix.
  3. Prove that the composition of transformations corresponds to the product of matrices and apply it to the composition of rotations and of reflections.
  4. Interpret the determinant as a factor of area and of orientation, decide when a transformation is invertible and express the rotation of axes of a second-degree equation with matrices.

Matrices as transformations

In the lesson of the course Algebra and Geometry I on vectors in the plane a vector was defined as an ordered pair of real numbers, and in the lesson on matrices and matrix operations the product of one matrix by another, row by column, was defined. If the pair (x,y)(x, y) is written as a column, that is, as a matrix with two rows and one column, a matrix AA with two rows and two columns can multiply it, and the result is another column with two entries, that is, another vector. In this way each matrix moves the points of the plane, and the algebra of matrices acquires a geometric meaning. From here on we identify the pair (x,y)(x, y) with the column with entries xx and yy, and we call e⃗1=(1,0)\vec{e}_1 = (1, 0) and e⃗2=(0,1)\vec{e}_2 = (0, 1) the standard basis vectors, so that every vector is written v⃗=xe⃗1+ye⃗2\vec{v} = x\vec{e}_1 + y\vec{e}_2.

A transformation of the plane is a function that assigns to each vector of the plane another vector of the plane. If AA is a matrix with two rows and two columns, the transformation with matrix AA is the one that assigns to each v⃗\vec{v} the product Av⃗A\vec{v}.
T(v⃗)=Av⃗=(abcd)(xy)=(ax+bycx+dy)T(\vec{v}) = A\vec{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}

For example, the matrix with entries 22, 00, 00 and 33 takes the point (x,y)(x, y) to the point (2x,3y)(2x, 3y): it stretches the plane to twice its size in the horizontal direction and to three times its size in the vertical one. Transformations with a matrix have a property that distinguishes them from the other functions of the plane, and which is expressed by the following definition.

A transformation TT of the plane is linear if it preserves the sum of vectors and the product of a vector by a number: for all u⃗\vec{u} and v⃗\vec{v} in the plane and every real number tt the two equalities below hold.
T(u⃗+v⃗)=T(u⃗)+T(v⃗),T(tv⃗)=tT(v⃗)T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v}), \qquad T(t\vec{v}) = tT(\vec{v})
TheoremThe columns of the matrix

Every transformation with a matrix is linear, and the columns of its matrix are the images of the standard basis vectors: the first is the image of e⃗1\vec{e}_1 and the second that of e⃗2\vec{e}_2. Conversely, every linear transformation of the plane is the transformation of a unique matrix, namely the one whose columns are T(e⃗1)T(\vec{e}_1) and T(e⃗2)T(\vec{e}_2).

We use the distributivity of the matrix product and its compatibility with multiplication by a number, proved in the lesson on matrices and matrix operations. The figure follows the transformation with matrix AA with columns (2,1)(2, 1) and (−1,1)(-1, 1).

Proof

  1. T(u⃗+v⃗)=A⋅(u⃗+v⃗)=Au⃗+Av⃗,T(tv⃗)=A⋅(tv⃗)=tAv⃗T(\vec{u} + \vec{v}) = \resaltar{A \cdot \left( \vec{u} + \vec{v} \right)} = \resaltar{A\vec{u} + A\vec{v}}, \qquad T(t\vec{v}) = \resaltar{A \cdot \left( t\vec{v} \right)} = \resaltar{tA\vec{v}}

    Let AA be a matrix and TT its transformation. Distributivity and compatibility with multiplication by a number give the two conditions of linearity; in the figure, the parallelogram formed by u⃗\vec{u} and v⃗\vec{v} is carried to the one formed by their images.

  2. T(e⃗1)=(abcd)(10)=(ac)T(\vec{e}_1) = \resaltar{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}} = \dato{c1}{\resaltar{\begin{pmatrix} a \\ c \end{pmatrix}}}

    We compute the image of e⃗1\vec{e}_1: when multiplying row by column, only the first entry of each row of AA counts.

  3. T(e⃗2)=(abcd)(01)=(bd)T(\vec{e}_2) = \resaltar{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix}} = \dato{c2}{\resaltar{\begin{pmatrix} b \\ d \end{pmatrix}}}

    In the same way, in the image of e⃗2\vec{e}_2 only the second entry of each row counts.

  4. T(v⃗)=T(xe⃗1+ye⃗2)=xT(e⃗1)+yT(e⃗2)=xAe⃗1+yAe⃗2=Av⃗T(\vec{v}) = \resaltar{T(x\vec{e}_1 + y\vec{e}_2)} = xT(\vec{e}_1) + yT(\vec{e}_2) = \resaltar{xA\vec{e}_1 + yA\vec{e}_2} = A\vec{v}

    Now let TT be any linear transformation, and let AA be the matrix whose columns are T(e⃗1)T(\vec{e}_1) and T(e⃗2)T(\vec{e}_2). We write v⃗=xe⃗1+ye⃗2\vec{v} = x\vec{e}_1 + y\vec{e}_2 and apply linearity; by the two steps above, T(e⃗1)=Ae⃗1T(\vec{e}_1) = A\vec{e}_1 and T(e⃗2)=Ae⃗2T(\vec{e}_2) = A\vec{e}_2. In the figure, the grid of the plane is carried to a grid of parallelograms.

  5. Be⃗1=T(e⃗1)=Ae⃗1,Be⃗2=T(e⃗2)=Ae⃗2→B=AB\vec{e}_1 = T(\vec{e}_1) = A\vec{e}_1, \quad B\vec{e}_2 = T(\vec{e}_2) = A\vec{e}_2 \rightarrow B = A

    If another matrix BB also satisfied T(v⃗)=Bv⃗T(\vec{v}) = B\vec{v} for every v⃗\vec{v}, the columns of BB, which are Be⃗1B\vec{e}_1 and Be⃗2B\vec{e}_2, would coincide with those of AA; the matrix is unique.

The theorem reduces the study of a linear transformation to that of two vectors: to know the matrix it suffices to know where e⃗1\vec{e}_1 and e⃗2\vec{e}_2 go, and to know what a matrix does it suffices to read its columns. Several geometric properties also follow from linearity. The origin stays fixed, because A0⃗=0⃗A\vec{0} = \vec{0}; that is why translations, which move the origin, are not linear. The line with vector equation p⃗+td⃗\vec{p} + t\vec{d}, studied in the Course I lesson on vectors, is carried to the set of the points Ap⃗+tAd⃗A\vec{p} + tA\vec{d}, which is a line if Ad⃗A\vec{d} is not zero and a single point if it is. In the same way, a line segment is carried to a line segment or to a point, parallel lines are carried to parallel lines or to points, and the square with vertices 0⃗\vec{0}, e⃗1\vec{e}_1, e⃗1+e⃗2\vec{e}_1 + \vec{e}_2 and e⃗2\vec{e}_2, which we shall call the unit square, is carried to the parallelogram, perhaps flattened, formed by the two columns of the matrix.