Matrix of Linear Xformations

Transcription

1 Matrix of Linear Xformations Theorem: If "L" is a linear transformation mapping R n into R m, there exists an mxn matrix "A" such that Lx Ax. This matrix is called the Standard Matrix for the Linear Transformation "L". x n x k e k k Lx L k n x k e k k n x k Le ( k ) Lx Le ( )Le ( )... L e n ( ) x A L( e )Le ( )... L e n ( ) Example A: Find the image of the vector, u under the transformation, Lx matrix, "A". 5, x and find the standard Page of

2 Lu L 5 5 A ( L( e )Le ( )) L L The transformation is a pure contraction. 5 Example B: Find the image of the vector, u, under the transformation, Lx and find the standard matrix, "A". x Lu L 5 A ( L( e )Le ( )) L L The transformation is a projection of x onto the x -axis. Page of

3 5 Example C: Find the image of the vector, u, under the transformation, Lx standard matrix, "A". x x and find the Lu L 5 5 A ( L( e )Le ( )) L L This is a reflection about the line x x. Vectors, u and L(u) Page of

4 5 Example D: Find the image of the vector, u, x under the transformation, Lx and find the standard matrix, "A". x Lu L 5 5 A ( L( e )Le ( )) L L Vectors, u and L(u) The transformation is a rotation CCWof 9 degrees of x. Page 4 of

5 5 Example E: Find the image of the vector, u, under the transformation, Lx standard matrix, "A". and find the x x Lu L 5 5 A ( L( e )Le ( )) L L Vectors, u and L(u) The transformation is a rotation CW of 9 degrees of x. Page 5 of

6 Example : "L" is a vertical shear transformation that maps e into e e but leaves the vector e unchanged. Find the Standard Matrix of "L". e e Le ( ) A [L( e ),L( e )] Le Let us see how the unit square depicted below is transformed under "L". x x Page 6 of

7 Examine transformation of the 4 vertices. Here is the picture of that vertical shear transformation of the unit square. x x Example : "L" is a horizontal shear transformation that maps e into e e + but leaves the vector e unchanged. Find the Standard Matrix of "L". Page 7 of

8 e e Le ( ) + Le A [L( e ),L( e )] Let us see how the unit square depicted below is transformed under "L". x x Page 8 of

9 Examine transformation of the 4 vertices x x y t () y s () y u y4 v t s, u, v, Page 9 of

10 Example 4: "L" rotates points about the origin thru " φ " radians CCW. Find the Standard Matrix of "L". Rotate e about the origin CCW an angle " φ". The length of the new vector is "", just like e, but its components are different. Since this new vector is a unit vector that makes an angle "φ" with the positive x cos( φ) -axis, this must be that vector: Le ( ) cos φ sin φ sin φ Rotate e about the origin CCW an angle " φ". The length of the new vector is "", just like e, but its components are different. Since this new vector is a π unit vector that makes an angle "φ + " with the positive x -axis, this must be that vector:. cos φ sin φ + + π π. Page of

11 cos φ + π cos π sin π cos φ sin φ sin φ sin φ Te ( ) + π cos π sin π sin φ + cos φ cos φ sin φ cos φ A [T( e ),T( e )] cos φ sin φ sin φ cos φ Let us see how the unit square depicted below is π transformed under "T" for φ. 4 x x Page of

12 A cos π 4 sin π 4 sin π 4 cos π 4 Examine transformation of the 4 vertices. Page of

13 x x Now we examine Linear Transformations from a Standard Matrix perspective. Recall the definitions for onto and one-to-one transformations. Definition: A mapping L: R n --->R m is said to be onto R m if each vector "b" in R m is the image of at least one vector "x" in R n. Definition: A mapping L: R n --->R m is said to be one-to-one if each vector "b" in R m is the image of at most one vector "x" in R n. Page of

14 Example 5: Determine if the transformation Lx x + x from R 4 into R 4 is onto, one-to-one, x + x x + x 4 both, or neither. A L( e )Le ( )Le ( )Le 4 ( ) This matrix has only pivots, and not 4. Therefore, the equation: Ax b has more than just the trivial solution, x. Accordingly, "L" is NOT one-to-one. Note that the matrix, "A", has 4 columns and only pivots. Its column vectors can not span R 4. Accordingly, "L" does not map R 4 onto R 4 and thus can NOT be onto. This mapping "T" is thus neither one-to-one nor onto. We proved the following theorem earlier, but it is worthy of repeating. Page 4 of

15 Theorem: Let L: R n --->R m be a Linear Transformation. Then "L" is one-to-one if and only if the equation Lx has only the trivial solution x. Let u v and both be in R n. If "L" is one-to-one, then Lu ( v) Lu Lv. Accordingly, L( x) has only the trivial solution. If Tl( x) has only the trivial solution, then Lu ( v) Lu Lv implies u v and thus that the vectors are not distinct. Otherwise, Lu Lv and thus "L" must be one-to-one. But this theorem is new, Theorem: Let L: R n --->R m be a Linear transformation and let "A" be the Standard Matrix for "L", then "L" maps R n onto R m if and only if the columns of "A" span R m ; "L" is one-to-one if and only if the columns of "A" are Linearly Independent. If the columns of "A" span R m, then the equation Ax b has at least one solution x in R n for every b in R m and thus "L" is onto. Page 5 of

16 If "L" is onto, then every x in R n has an image b in R m which implies that the equation Ax b is consistent for every b in R m and thus the columns of "A" must span R m. If the columns of "A" are Linearly Independent, then m nand every b in R m can be represented as a unique linear combination of the columns of "A" and so distinct vectors in R n have distinct images in R m. Thus, "L" is one-to-one, If "L" is one-to-one, then distinct vectors x in R n produce distinct vectors b in R m under "L". Accordingly, Ax b has a unique solution for each b. This in turn requires that the columns of "A" be linearly independent. Page 6 of

17 Example 6:Determine if the transformation x + 5x + 7x Lx x + x from R 4 into R 4 is onto, x one-to-one, both, or neither. A L( e )Le ( )Le ( ) 5 7 The columns of "A" span R ; therefore, the mapping is onto. The columns of "A" are linearly independent; therefore, the mapping is also one-to-one. Theorem: If E v, v,..., v n and F w, w,..., w n are ordered bases for "V" & "W", then for each Linear Transformation "L" from "V" to "W", there is an mxn matrix "A" such that L( v) F A( v) E. v c v + c v c n v n F F F Lv F c Lv + c Lv C n Lv n Page 7 of

18 T ( v) E c, c... c n ( F Lv ( ) F... L( v n ) F ) A L v Example 7: Let b, b, and b and let "L" be a linear transformation that takes R intor such that L( x) x b + x b + ( x + x ) b. Find the matrix "A" representing "L" with respect to the bases ( e e ) & ( b b b ). b Lx x b + x b + x + x b Le b Le + b + b + b + b β Le Le β Page 8 of

19 ( F Lv ( ) F... L( v n ) F ) A L v Le ( β Le ( ) β ) A Theorem: If "A" is the matrix representing the linear transformation "L" that takes R n into R m with respect to the bases E ( u, u,..., u n ) and F ( b, b,..., b m ), then the reduced row echelon form of the matrix "B" augmented by the ordered columns of L( u k ), where B ( b b...b m ), is the identity matrix augmented by the ordered columns of "A". The matrix ( B I L( u )Lu ( )... L u n equivalent to B ( B I L( u )Lu ( )... L u n B ( B I L( u )Lu ( )... L( u n ) ) ( I I B Lu ( ) B Lu ( )...B Lu ( n ) ) ( I I a a... a n ) ( I I A ). ) is row ). Page 9 of

20 Example 8: Let E u, u, u and F b, b, where u, u, u and b, b. For the transformation Lx x x from R into R, find the matrix "A" representing "L" with respect to the ordered bases "E" & "F". Lu ( ) BA Lu ( ) Lu Lu ( )Lu Lu B ( b b ) B A B ( ) Lu Lu ( )Lu A Page of

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