projmatrix - Introduction to Linear Transformations...

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Introduction to Linear Transformations Projection Example: Let ± v = ± 1 , 3 ² . The vector projection of a vector ± x = ± x 1 ,x 2 ² in the direction of ± v is given by: proj ± v ± x = ± x ± v ± v ± v ± v Expand out the dot product: proj ± v ± x = x 1 +3 x 2 10 ± 1 , 3 ² = ³ 1 10 x 1 + 3 10 x 2 , 3 10 x 1 + 9 10 x 2 ´ We can write this expression in a very useful way using matrix notation . First of all, let’s write the vectors with the coordinates arranged vertically instead of horizontally. So, instead of writing ± v = ± 1 , 3 ² and ± x = ± x 1 2 ² , we will use the notation: ± v = µ 1 3 ± x = µ x 1 x 2 Each of these vertical arrangements have two rows and one column and will be referred to as a two by one matrix. In this notation, the projection operation would be: proj ± v ± x = 1 10 x 1 + 3 10 x 2 3 10 x 1 + 9 10 x 2 A matrix can have more than one column. The following would be called a two by two matrix: A = µ a 11 a 12 a 21 a 22 So, for example, the entries of the following matrix displays the coe±cients of the coordi- nates of the projection we have just considered: A = µ 1 / 10 3 / 10 3 / 10 9 / 10
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Let us defne the expression A ± x (multiplication oF a vector by a matrix) according to the Following Formula: A ± x = ± a 11 a 12 a 21 a 22 ²± x 1 x 2 ² = ± a 11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 ² It Follows immediately that our projection operation can be written as: proj ± v ± x = ± 1 / 10 3 / 10 3 / 10 9 / 10 x 1 x 2 ² We can describe the coordinates oF A ± x in terms oF dot products. IF we take the dot product oF the frst row oF A with the vector ± x , we will produce the frst coordinate oF A ± x . Similarly, iF we take the dot product oF the second row oF A with the vector ± x ,we will produce the second coordinate oF A ± x . Incidentally, we can write the matrix in a more compact Form iF we “factor out” the 1 10 . That is, proj ± v ± x = ± 1 / 10 3 / 10 3 / 10 9 / 10 x 1 x 2 ² = 1 10 ± 13 39 x 1 x 2 ² More generally, let’s defne the multiplication oF a matrix by a number c by the Following Formula: c A = c ± a 11 a 12 a 21 a 22 ² = ± ca 11 ca 12 ca 21 ca 22 ² This should remind you oF how we multiply vectors by scalars. Linearity of the Projection The projection in the preceding example can be described as a vector-valued Function oF a vector variable ± x . It has an important property called linearity . Let’s defne some specifc vectors ± x , ± y and ± z as Follows: ± x = ± 1 1 ² ± y = ± 2 1 ² ± z = ± 3 2 ² Please note that ± z = ± x + ± y . Now, consider their vector projections in the direction oF ± v . proj ± v ± x = 1 10 ± 1 1 ² = 1 10 ± 4 12 ² proj ± v ± y = 1 10 ± 2 1 ² = 1 10 ± 5 15 ² proj ± v ± z = 1 10 ± 3 2 ² = 1 10 ± 9 27 ² Notice that proj ± v ± z =proj ± v ± x +pro j ± v ± y . In other words: proj ± v ( ± x + ± y )=proj ± v ± x j ± v ± y
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To see why this is true, remember that the vector sum ± x + ± y can be represented diagram- matically by the parallelogram rule: We can also represent this by a vector triangle where the head of ± y meets the tail of ± x Now, project these vectors onto ±
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This note was uploaded on 03/20/2011 for the course MA 243 taught by Professor Poon during the Spring '09 term at Embry-Riddle FL/AZ.

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projmatrix - Introduction to Linear Transformations...

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