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< 14.1 Basics of Linear Algebra | Contents | 14.3 Systems of Linear Equations >
Linear Transformations¶
For vectors \(x\) and \(y\), and scalars \(a\) and \(b\), it is sufficient to say that a function, \(F\), is a linear transformation if
It can be shown that multiplying an \({m} \times {n}\) matrix, \(A\), and an \({n} \times {1}\) vector, \(v\), of compatible size is a linear transformation of \(v\). Therefore from this point forward, a matrix will be synonymous with a linear transformation function.
TRY IT! Let \(x\) be a vector and let \(F(x)\) be defined by \(F(x) = Ax\) where \(A\) is a rectangular matrix of appropriate size. Show that \(F(x)\) is a linear transformation.
Proof: Since \(F(x) = Ax\), then for vectors \(v\) and \(w\), and scalars \(a\) and \(b\), \(F(av + bw) = A(av + bw)\) (by definition of \(F\))\(=\)\(aAv + bAw\) (by distributive property of matrix multiplication)\(=\)\(aF(v) + bF(w)\) (by definition of \(F\)).
If \(A\) is an \({m} \times {n}\) matrix, then there are two important subpsaces associated with \(A\), one is \({\mathbb{R}}^n\), the other is \({\mathbb{R}}^m\). The domain of \(A\) is a subspace of \({\mathbb{R}}^n\). It is the set of all vectors that can be multiplied by \(A\) on the right. The range of \(A\) is a subspace of \({\mathbb{R}}^m\). It is the set of all vectors \(y\) such that \(y=Ax\). It can be denoted as \(\mathcal{R}(\mathbf{A})\), where \(\mathcal{R}(\mathbf{A}) = \{y \in {\mathbb{R}}^m: Ax = y\}\). Another way to think about the range of \(A\) is the set of all linear combinations of the columns in \(A\), where \(x_i\) is the coefficient of the ith column in \(A\). The null space of \(A\), defined as \(\mathcal{N}(\mathbf{A}) = \{x \in {\mathbb{R}}^n: Ax = 0_m\}\), is the subset of vectors in the domain of \(A, x\), such that \(Ax = 0_m\), where \(0_m\) is the zero vector (i.e., a vector in \({\mathbb{R}}^m\) with all zeros).
TRY IT! Let \(A = [[1, 0, 0], [0, 1, 0], [0, 0, 0]]\) and let the domain of \(A\) be \({\mathbb{R}}^3\). Characterize the range and nullspace of \(A\).
Let \(v = [x,y,z]\) be a vector in \({\mathbb{R}}^3\). Then \(u = Av\) is the vector \(u = [x,y,0]\). Since \(x,y\in {\mathbb{R}}\), the range of \(A\) is the \(x\)-\(y\) plane at \(z = 0\).
Let \(v = [0,0,z]\) for \(z\in {\mathbb{R}}\). Then \(u = Av\) is the vector \(u = [0, 0, 0]\). Therefore, the nullspace of \(A\) is the \(z\)-axis (i.e., the set of vectors \([0,0,z]\) \(z\in {\mathbb{R}}\)).
Therefore, this linear transformation “flattens” any \(z\)-component from a vector.
< 14.1 Basics of Linear Algebra | Contents | 14.3 Systems of Linear Equations >