index.md (1927B)
1 +++ 2 title = 'Linear transformations' 3 template = 'page-math.html' 4 +++ 5 6 # Linear transformations 7 definitions: 8 * transformation, function, mapping: rule assigning to each vector in $\Re^n$ a vector $T(x)$ in $\Re^m$ 9 * domain: set $\Re^n$ 10 * codomain: set $\Re^m$ 11 * image: vector T(x) 12 * range: set of all images T(x) 13 14 a projection transformation happens if you go to a lower dimension (e.g. $x_3$ becomes 0). a shear transformation happens if a 2D square is tilted sideways into a parallelogram. 15 16 a transformation T is linear if: 17 i) $T(u + v) = T(u) + T(v)$ for all $u,v \in \text{Domain}(T)$ 18 ii) $T(cu) = cT(u)$ for all scalars c and all $u \in \text{Domain}(T)$ 19 20 linear transformations preserve operations of vector addition and scalar multiplication. 21 22 if T is a linear transformation, then: 23 * $T(0) = 0)$ 24 * $T(cu + dv) = cT(u) + dT(v)$ 25 * $T(c_1 v_2 + \dots + c_p v_p) = c_1 T(v_1) + \dots + c_p T(v_p)$ (superposition principle) 26 27 given scalar r, and $T: \Re^2 \rightarrow \Re^2$ by $T(x) = rx$ 28 * contraction: when $0 \leq r \leq 1$ 29 * dilation: when $r > 1$ 30 31 every linear transformation $\Re^n \rightarrow \Re^m$ is a matrix transformation $x \mapsto Ax$. 32 33 $A = [[T(e_1) \dots T(e_n)]$, where $e_j$ is the jth column of the identity matrix in $\Re^n$ 34 35 geometric linear transformations of $\Re^2$: 36 37    38 39 types of mappings: 40 * $T: \Re^n \rightarrow \Re^m$ is 'onto' $\Re^m$ if _each_ b in $\Re^m$ is the image of _at least one_ x in $\Re^n$. 41 * $T: \Re^n \rightarrow \Re^m$ is one-to-one if _each_ b in $\Re^m$ is the image of _max one_ x in $\Re^n$. 42 * so if $T(x) = 0$ only has the trivial solution 43 44 for mapping $T: \Re^n \rightarrow \Re^m$ and standard matrix $A$: 45 * T maps $\Re^n$ onto $\Re^m$ iff columns of matrix span $\Re^m$ 46 * T is one-to-one iff columns of matrix are linearly independent. 47