structures:matrix:linear
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structures:matrix:linear [2025/03/03 11:08] simon |
structures:matrix:linear [2025/03/13 08:00] (current) simon [linear transformations with matrices] |
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| - | ====Linear==== | + | =====Linear===== |
| - | This is an idea that crops up in many contexts. Starting from //Linear Relationships// which was a big topic year 9 in school. That is the kind of relationship between 2 values that shows up as a straight line on a graph in cartesian space with those values as the axes. Then extended to many variables. | + | This is an idea that crops up in many contexts. Starting from **[[#Linear Relationships]]** which was a big topic year 9 in school. |
| - | The equation \(\ ax+by+cz=d\ \) describes a linear relationship between `x`, `y` and `z` when `a`, `b`, `c` and `d` are constants. This relationship could be shown as a plane in a 3D graph. A Linear Relationship graphs as a __flat__ shape one dimension smaller than the number of variables. Some of the constants might be 0, you will still get a plane in 3D space (or nothing or everywhere if \(a=b=c=0\!\) ). | + | ====Linear Relationships==== |
| + | That is the kind of relationship between 2 values that shows up as a straight line on a graph in cartesian space with those values as the axes. Then extended to many variables. The equation \(\ ax+by+cz=d,\ \) when `a`, `b`, `c` and `d` are constants, describes a linear relationship between `x`, `y` and `z.` This relationship could be shown as a plane in a 3D graph. A Linear Relationship graphs as a __flat__ space one dimension smaller than the number of variables. Some of the constants might be 0, you will still get a plane in 3D space (or nothing or everywhere if \(a=b=c=0\!\) ). | ||
| - | ===Linear Combination=== | + | The statement "`y` is proportional to `x`" is a stronger, saying \(\ y=kx\ \) with `k` constant. In this case the graph is a line through the origin. Expressed as a function, \(\ {\rm f}(x)=kx,\ \) then \(\rm f\) is a **[[#Linear Map]]**, also written \(\ x\mapsto kx.\) |
| + | ====Linear Combination==== | ||
| \(\ ax+by+cz\ \) is a //Linear Combination// of three variables `x, y` and `z`. Much more generally a //Linear Combination// of a set of things is made by multiplying each by a scalar (possibly zero or one) and adding the results. If there are `n` elements in the set `X` then we could write this \(\ \sum_{i=1}^n a_i x_i\ \) where `a_i` are scalars and `x_i` are the elements of `X`. | \(\ ax+by+cz\ \) is a //Linear Combination// of three variables `x, y` and `z`. Much more generally a //Linear Combination// of a set of things is made by multiplying each by a scalar (possibly zero or one) and adding the results. If there are `n` elements in the set `X` then we could write this \(\ \sum_{i=1}^n a_i x_i\ \) where `a_i` are scalars and `x_i` are the elements of `X`. | ||
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| Matrices have a great deal to do with that. | Matrices have a great deal to do with that. | ||
| - | ===Linear map=== | + | ====Linear Map==== |
| is from space to space preserving addition and scalar multiplication. By this we mean we can do the transformation then the operation __or__ do the operation first then do the transformation and the result will be the same. | is from space to space preserving addition and scalar multiplication. By this we mean we can do the transformation then the operation __or__ do the operation first then do the transformation and the result will be the same. | ||
| \[{\rm f}:V\mapsto W\quad\text{is linear iff}\quad{\rm f}({\bf a}+{\bf b})={\rm f}({\bf a})+{\rm f}({\bf b})\quad\text{and}\quad{\rm f}(c{\bf a})=c{\rm f}({\bf a}),\] | \[{\rm f}:V\mapsto W\quad\text{is linear iff}\quad{\rm f}({\bf a}+{\bf b})={\rm f}({\bf a})+{\rm f}({\bf b})\quad\text{and}\quad{\rm f}(c{\bf a})=c{\rm f}({\bf a}),\] | ||
| so linear combinations are preserved (since addition and scalar multiplication each associative). | so linear combinations are preserved (since addition and scalar multiplication each associative). | ||
| - | eg \(\ {\bf v}\mapsto c{\bf v}\ \) is linear. But \(\ x\mapsto x^2\ \) and \(\ x\mapsto x+1\ \) are not linear. The last is //affine// instead. For a cartesian or vector space any rotation, reflection, stretching or skewing which preserves the origin is a linear map, a linear transformation. For the appropriate space of [[discussion#functions]] differentiation, definite integral and indefinite integral with a fixed starting point are linear maps, linear operations on those functions producing new functions. Expected value of a random variable is a linear function (but variance is not). I've seen HSC questions that are very quickly answered if you understand the meaning of this last sentence, though most students just use particular memorised properties from the 'expected value' topic --- properties that are actually true of __any__ linear function, and much much more useful than that single, specific, topic. Many probably could not name or use the much more general and widespread idea and think of the topic in isolation as one more group of methods to memorise. | + | eg \(\ {\bf v}\mapsto c{\bf v}\ \) is linear. But \(\ x\mapsto x^2\ \) and \(\ x\mapsto x+1\ \) are not linear. The last is //affine// instead. For a cartesian or vector space any rotation, reflection, stretching or skewing is a linear map, a linear transformation, if it preserves the origin. That is if it is around, across or through the origin. the For the appropriate space of [[discussion#functions]] differentiation, definite integral and indefinite integral with a fixed starting point are linear maps, linear operations on those functions producing new functions. Expected value of a random variable is a linear function (but variance is not). I've seen HSC questions that are very quickly answered if you understand the meaning of this last sentence, though most students just use particular memorised properties from the 'expected value' topic --- properties that are actually true of __any__ linear function, and much much more useful than that single, specific, topic. Many probably could not name or use the much more general and widespread idea and think of the topic in isolation as one more group of methods to memorise. |
| We can use [[discussion#matrix]] algebra and notation to work with transformations including linear and affine ones. | We can use [[discussion#matrix]] algebra and notation to work with transformations including linear and affine ones. | ||
| - | ===linear transformations with matrices=== | + | ====Linear transformations with matrices==== |
| Now consider [[applications#square matrices]]. For matrix multiplication with `n`-dimenional square matrices we have an //Identity//, the //Unit// for multiplication, like `1` in the scalars, | Now consider [[applications#square matrices]]. For matrix multiplication with `n`-dimenional square matrices we have an //Identity//, the //Unit// for multiplication, like `1` in the scalars, | ||
| \[%%I=\left[\matrix{ | \[%%I=\left[\matrix{ | ||
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| \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) Scale by 3 around origin\\ \\ | \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) Scale by 3 around origin\\ \\ | ||
| \(\left[\matrix{0 & -1 \cr 1 & 0 }\right],\quad\) Rotate right angle anticlockwise around origin\\ \\ | \(\left[\matrix{0 & -1 \cr 1 & 0 }\right],\quad\) Rotate right angle anticlockwise around origin\\ \\ | ||
| - | \(\left[\matrix{1 & 0 \cr 0 & -1 }\right],\quad\) Reflect on horizontal axis, negate vertical axis\\ \\ | + | \(\left[\matrix{1 & 0 \cr 0 & -1 }\right],\quad\) Reflect on horizontal axis (negate vertical axis)\\ \\ |
| - | \(\left[\matrix{-1 & 0 \cr 0 & -1 }\right],\quad\) Rotate half circle around origin, reflect vert then horiz\\ \\ | + | \(\left[\matrix{-1 & 0 \cr 0 & -1 }\right],\quad\) Rotate half circle around origin (reflect vert then horiz)\\ \\ |
| - | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis\\ \\ | + | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis (zero the horiz component)\\ \\ |
| - | \(\left[\matrix{1 & 2 \cr 0 & 1 }\right],\quad\) Horizontal shear\\ \\ | + | \(\left[\matrix{1 & \frac12 \cr 0 & 1 }\right],\quad\) Horizontal shear (add half vert component to horiz one)\\ \\ |
| \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | ||
| \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | ||
| Continued [[discussion#complex-as-matrix|in the main discussion]]. | Continued [[discussion#complex-as-matrix|in the main discussion]]. | ||
structures/matrix/linear.1740960513.txt.gz · Last modified: 2025/03/03 11:08 by simon
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