Fundamentals of Matrix Algebra, 2011a

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Chapter 5 Graphical Exploraons of Vectors Diagonal reflecon across the line y = x. [ 0 ] 1 . x 1 0 y y Rotaon around the origin by an angle of θ. [ ] cos θ − sin θ sin θ cos θ θ . x Projecon onto the x axis. (Note how the square is “squashed” down onto the x-axis.) [ ] 1 0 0 0 y . x Projecon onto the y axis. (Note how the square is “squashed” over onto the y-axis.) [ ] 0 0 0 1 y . x Now that we have seen a healthy list of transformaons that we can perform on the Cartesian plane, let’s pracce a few more mes creang the matrix that gives the desired transformaon. In the following example, we develop our understanding one 198
5.1 Transformaons of the Cartesian Plane more crical step. . Example 97 .Find the matrix A that transforms the Cartesian plane by performing the following operaons in order: 3. Horizontal stretch by a factor 1. Vercal shear by a factor of of θ = 30 ◦ 0.5 2. Counterclockwise rotaon of 2 4. Diagonal reflecon across about the origin by an angle the line y = x S Wow! We already know how to do this – sort of. We know we can find the columns of A by tracing where ⃗e 1 and ⃗e 2 end up, but this also seems difficult. There is so much that is going on. Fortunately, we can accomplish what we need without much difficulty by being systemac. First, let’s perform the vercal shear. The matrix that performs this is A 1 = [ ] 1 0 . 0.5 1 Aer that, we want to rotate everything clockwise by 30 ◦ . To do this, we use A 2 = [ ] cos 30 ◦ − sin 30 ◦ sin 30 ◦ cos 30 ◦ = [ √ 3/2 −1/2 1/2 √ 3/2 ] . In order to do both of these operaons, in order, we mulply A 2 A 1 . 8 To perform the final two operaons, we note that A 3 = [ ] 2 0 0 1 and A 4 = [ ] 0 1 1 0 perform the horizontal stretch and diagonal reflecon, respecvely. Thus to perform all of the operaons “at once,” we need to mulply by A = A 4 A 3 A 2 A 1 [ ] [ ] [ √ ] [ ] 0 1 2 0 3/2 = √ −1/2 1 0 1 0 0 1 1/2 3/2 0.5 1 [ √ √ ] ( 3 + 2)/4 3/2 = (2 √ 3 − 1)/2 −1 [ ] 0.933 0.866 ≈ . 1.232 −1 8 The reader might ask, “Is it important to do mulply these in that order? Could we have mulplied A 1 A 2 instead?” Our answer starts with “Is matrix mulplicaon commutave?” The answer to our queson is “No,” so the answers to the reader’s quesons are “Yes” and “No,” respecvely. 199
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. . . Fundamentals of Matrix Algebr
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Copyright © 2011 Gregory Hartman L
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P A Note to Students, Teachers, and
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Contents 5 Graphical Exploraons of
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Chapter 1 Systems of Linear Equaons
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Chapter 2 Matrix Arithmec In the pa
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Chapter 2 Matrix Arithmec ⎡ 3 6
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Chapter 2 Matrix Arithmec by the nu
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Chapter 2 Matrix Arithmec means? Yo
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Chapter 2 . Matrix Arithmec ⎡ ⎤
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Chapter 2 Matrix Arithmec using the
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Chapter 2 Matrix Arithmec the numbe
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Chapter 2 Matrix Arithmec BB = BC,
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Chapter 2 Matrix Arithmec identy ma
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Chapter 2 Matrix Arithmec (a) Give
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Chapter 2 Matrix Arithmec 2.3 Visua
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Chapter 2 Matrix Arithmec Vector Ad
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Chapter 2 Matrix Arithmec Scalar Mu
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Chapter 2 Matrix Arithmec S To draw
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Chapter 2 Matrix Arithmec . Ḋefin
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Chapter 2 Matrix Arithmec y A⃗x A
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Chapter 2 Matrix Arithmec Of course
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Chapter 2 Matrix Arithmec 2.4 Vecto
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Chapter 2 Matrix Arithmec In previo
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Chapter 2 Matrix Arithmec Again, In
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Chapter 2 Matrix Arithmec equaon is
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Chapter 2 Matrix Arithmec “x 3
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Chapter 2 Matrix Arithmec . Example
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Chapter 2 Matrix Arithmec [ ] 1 −
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Chapter 2 Matrix Arithmec We know h
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Chapter 2 Matrix Arithmec Noce also
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Chapter 2 Matrix Arithmec 4. T/F: I
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Chapter 2 Matrix Arithmec Since mat
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Chapter 2 Matrix Arithmec We have a
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Chapter 2 Matrix Arithmec . Ṫheor
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Chapter 2 Matrix Arithmec ⎡ −15
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Chapter 2 Matrix Arithmec is that t
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Chapter 2 Matrix Arithmec We now ap
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Chapter 2 Matrix Arithmec more to b
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Chapter 2 Matrix Arithmec can cause
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Chapter 3 Operaons on Matrices .
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Chapter 3 Operaons on Matrices Ther
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Chapter 3 Operaons on Matrices Find
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Chapter 3 Operaons on Matrices Let
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Chapter 3 Operaons on Matrices oper
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Chapter 3 Operaons on Matrices The
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Chapter 3 Operaons on Matrices We e
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Chapter 3 Operaons on Matrices In t
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Chapter 3 Operaons on Matrices S To
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Chapter 3 Operaons on Matrices We
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices C 1,
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Chapter 3 Operaons on Matrices 21.
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Page 245 and 246: (d) -1 (e) 0 ⎡ 5. (a) λ 1 = −4
Page 247 and 248: Index ansymmetric, 128 augmented ma
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Fundamentals of Matrix Algebra, 2011a

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