Transformations of Spans and Loci

Section 7.2 Transformations of Spans and Loci

Subsection 7.2.1 Transformations of Spans

Now that I understand how matrices describe transformations, I want to know how transformations affect linear subspaces. I know that they preserve them (and each linear subspace is sent to some other linear subspace), but I want to be more specific: given a particular linear subspace, can I determine exactly where it goes under a transformation?

Definition 7.2.1.

If \(M: \RR^n \rightarrow \RR^m\) is a transformation (represented by a \(m \times n\) matrix \(M\)) and \(L\) is a linear or affine subspace of \(\RR^n\text{,}\) then the result of the matrix \(M\) acting on all vectors in \(L\) will be a linear or affine subspace of \(\RR^m\) called the image of \(L\) under \(M\). The image of all of \(\RR^n\) under \(M\) is simply called the image of \(M\).

The most important property of linear transformations is that they preserve addition and scalar multiplication. That means they preserve the linear combinations. Let \(a_i\) be scalars and \(v_i\) be vectors in \(\RR^n\text{,}\) with \(M\) a \(m\times n\) matrix.

\begin{align*} M(a_1 v_1 + a_2v_2 + \ldots + a_kv_k) \amp = M(a_1v_1) + M(a_2v_2) + \ldots + M(a_kv_k)\\ \amp = a_1(Mv_1) + a_2(Mv_2) + \ldots + a_k (Mv_k) \end{align*}

This calculation shows that the image of a linear combination is still a linear combination, just in the vectors \(Mv_i\) instead of \(v_i\text{.}\) Since all elements of a span are linear combinations, everything in the span of the \(v_i\) is sent to the span of the vectors \(Mv_i\text{.}\) I can summarize these observations in a proposition.

Proposition 7.2.2.

A linear transformation represented by a \(m \times n\) matrix \(M\) sends spans to spans. In particular, the span \(\Span \{ v_1, v_2, \ldots, v_k\}\) is sent to \(\Span \{ Mv_1, Mv_2, \ldots, Mv_k\}\text{.}\)

Matrices acting on spans are easy: I just figure out where the individual vectors go and the span of those vectors will be transformed into the span of their images. Offset spans are almost as easy: consider an affine subspace \(u + \Span \{ v_1, v_2, \ldots, v_k\}\text{.}\) Any element of this looks like \(u + a_1v_1 + a_2v_2 + \ldots a_kv_k\text{.}\) I can calculate where this linear combination goes under the action of \(M\text{.}\)

\begin{equation*} M(u + a_1v_1 + a_2v_2 + \ldots + a_kv_k) = Mu + a_1 Mv_1 + a_2Mv_2 + \ldots + a_kMv_k \end{equation*}

The offset span is still sent to an offset span, with offset \(Mu\) and spanning vectors \(Mv_i\text{.}\)

Be careful that the matrix need not preserve the dimension of the span. Even if the \(v_i\) are linearly independent and form a basis for the span, the vectors \(Mv_i\) may not be linearly independent. The dimension of the new span might be smaller. Linear combinations are sent to linear combinations, but linear independence may not be preserved.

Subsection 7.2.2 Transformation of Loci

For loci, the picture is much more complicated. Equations do not transform nearly as pleasantly as spans. Planes in \(\RR^3\) are defined by a normal; I might hope that the new plane is defined by the image of the normal. Unfortunately, since the matrix may not preserve orthogonality, this will usually not happen. To determine the image of a locus, the best approach is to describe the locus as a span or offset span to find the image of that span.

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