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projmatrix - Introduction to Linear Transformations...

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Introduction to Linear Transformations Projection Example: Let v = 1 , 3 . The vector projection of a vector x = x 1 , x 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 = 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 and x = x 1 , x 2 , we will use the notation: v = 1 3 x = 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 = a 11 a 12 a 21 a 22 So, for example, the entries of the following matrix displays the coe ffi cients of the coordi- nates of the projection we have just considered: A = 1 / 10 3 / 10 3 / 10 9 / 10
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Let us define the expression Ax (multiplication of a vector by a matrix) according to the following formula: Ax = a 11 a 12 a 21 a 22 x 1 x 2 = a 11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 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 We can describe the coordinates of Ax in terms of dot products. If we take the dot product of the first row of A with the vector x , we will produce the first coordinate of Ax . Similarly, if we take the dot product of the second row of A with the vector x , we will produce the second coordinate of Ax . 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 = 1 10 1 3 3 9 x 1 x 2 More generally, let’s define the multiplication of a matrix by a number c by the following formula: c A = c a 11 a 12 a 21 a 22 = ca 11 ca 12 ca 21 ca 22 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 define some specific vectors x , y and z as follows: x = 1 1 y = 2 1 z = 3 2 Please note that z = x + y . Now, consider their vector projections in the direction of v . proj v x = 1 10 1 3 3 9 1 1 = 1 10 4 12 proj v y = 1 10 1 3 3 9 2 1 = 1 10 5 15 proj v z = 1 10 1 3 3 9 3 2 = 1 10 9 27 Notice that proj v z = proj v x + proj v y . In other words: proj v ( x + y ) = proj v x + proj 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 v The arrow in red is proj v y and the arrow in green is proj v x . The head of the red arrow meets the tail of the green arrow so the combination of the red and green arrows is the sum proj v x + proj v y . However, if we drop a perpendicular from z to v , we see that proj v x + proj v y is exactly the same as proj v z .
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