Instructor's Solution Manual for Elementary Statistics: Picturing the World 7th Edition by Ron Larson, Chapter 1-11 | All Chapters

SOLUTION MANUAL
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,Contents


1 Linear Algebra and Optimization: An Introduction
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2 Linear Transformations and Linear Systems
U U U U 17


3 Diagonalizable Matrices and Eigenvectors U U U 35


4 OptimizationBasics:AMachineLearningView U U U U U 47


5 Optimization Challenges and Advanced Solutions
U U U U 57


6 Lagrangian Relaxation and Duality
U U U 63


7 Singular Value Decomposition
U U 71


8 Matrix Factorization
U 81


9 The Linear Algebra of Similarity
U U U U 89


10 The Linear Algebra of Graphs
U U U U 95


11 Optimization in Computational GraphsU U U 101

,Chapter 1 U




Linear Algebra and Optimization: An Introduction
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1. For any two vectors x and y, which are each of length a, show that (i)
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x− y is orthogonal tox+y, and(ii) the dot product of x−3y and x+3y is
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negative.
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(i) The first is simp·ly−x x y y using the distributive property of matrix
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multiplication. The d· ot product of a vector with itself is its squared length.
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U



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Sincebothvectorsareofthesamelength,itfollowsthattheresultis0.(ii)
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In the second case, one can use a similar argument to show that the result is a2 −
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U




9a2, which is negative.
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2. Consider a situation in which you have three matrices A, B, and C, of sizes U U U U U U U U U U U U U U




10×2, 2×10,and 10×10, respectively. U U U U




(a) Suppose you had to compute the matrix product ABC. From an efficiency per- U U U U U U U U U U U U




spective, wouldit computationally makemore sensetocompute (AB)C or would
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itmakemoresensetocomputeA(BC)?
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(b) If you had to compute the matrix product CAB, would it make more sense to
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compute (CA)B or C(AB)? U U U U




The main point is to keep the size of the intermediate matrix as small as
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possible in order to reduce both computational and space requirements. In
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the case of ABC, it makes sense to compute BC first. In the case of CAB it
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makes sense to compute CA first. This type of associativity property
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isused frequently inmachinelearningin orderto reducecomputational
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requirements.
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3. Show that if a matrix A satisfies A = AT,then allthe diagonal elements
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of thematrixare0.
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Note that A + AT = 0. However, this matrix also contains twice the
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diagonal elements of A on its diagonal. Therefore, the diagonal
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elementsofAmustbe0.
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4. Show that ifwe have a matrix satisfying A= UAT,thenforanycolumn U U U U U U U U U U




vectorx,wehave x Ax=0.
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T
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U



U U




Note that the transpose of the scalar xT Ax remains unchanged. Therefore,
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U



U U U




1

, we have
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xTAx=(xTAx)T =xTATx=−xTAx. Therefore, we have 2xTAx=0.
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2