Kenneth Kuttler

18.4. SUBSPACES SPANS AND BASES 397

47. Use the result of Problem 46 to verify directly that (AB)T = BT AT without makingany reference to subscripts.

48. Suppose A is an n×n matrix and for each j,

∑i=1

∣∣Ai j∣∣< 1

Show that the infinite series ∑∞k=0 Ak converges in the sense that the i jth entry of the

partial sums converge for each i j. Hint: Let R ≡ max j ∑ni=1

∣∣Ai j∣∣ . Thus R < 1. Show

that ∣∣∣∣∣∑i

(A2)

i j

∣∣∣∣∣≤ R2.

Then generalize to show that∣∣∣∑i (Am)i j

∣∣∣ ≤ Rm. Use this to show that the i jth entryof the partial sums is a Cauchy sequence. From calculus, these converge by com-pleteness of the real or complex numbers. Next show that (I −A)−1 = ∑

∞k=0 Ak. The

Leontief model in economics involves solving an equation for x of the form

x= Ax+b, or (I −A)x= b

The vector Ax is called the intermediate demand and the vectors Akx have economicmeaning. From the above,

x= Ib+Ab+A2b+ · · ·

The series is also called the Neuman series. It is important in functional analysis.

49. Let a be a fixed vector. The function Ta defined by Tav = a+v has the effect oftranslating all vectors by adding a. Show this is not a linear transformation. Explainwhy it is not possible to realize Ta in R3 by multiplying by a 3×3 matrix.

50. In spite of Problem 49 we can represent both linear transformations and translationsby matrix multiplication at the expense of using higher dimensions. This is done bythe homogeneous coordinates. I will illustrate in R3 where most interest in this isfound. For each vector v = (v1,v2,v3)

T , consider the vector in R4 (v1,v2,v3,1)T .

What happens when you do1 0 0 a10 1 0 a20 0 1 a30 0 0 1



v1v2v31

?

Describe how to consider both linear transformations and translations all at once byforming appropriate 4×4 matrices.

18.4 Subspaces Spans and BasesThe span of some vectors consists of all linear combinations of these vectors. As explainedearlier, a linear combination of vectors is just a finite sum of scalars times vectors.

18.4. SUBSPACES SPANS AND BASES 39747. Use the result of Problem 46 to verify directly that (AB)’ = B’A™ without makingany reference to subscripts.48. Suppose A is ann Xn matrix and for each J,ny |Aij| <1i=lShow that the infinite series Y7°_)A* converges in the sense that the ij’” entry of thepartial sums converge for each ij. Hint: Let R= max; Y7_, A; (| . Thus R < 1. ShowthatY (4’);; <P’.lThen generalize to show that Dy (A™);;| < R”. Use this to show that the i j'” entryof the partial sums is a Cauchy sequence. From calculus, these converge by com-pleteness of the real or complex numbers. Next show that (J —A)! = Yeo A* . TheLeontief model in economics involves solving an equation for x of the formz=Ax+b,or J—-A)x=bThe vector Az is called the intermediate demand and the vectors A‘ have economicmeaning. From the above,z=Ib+Ab+A°b+---The series is also called the Neuman series. It is important in functional analysis.49. Let a be a fixed vector. The function 7, defined by T7,v = a+ v has the effect oftranslating all vectors by adding a. Show this is not a linear transformation. Explainwhy it is not possible to realize 7, in R* by multiplying by a 3 x 3 matrix.50. In spite of Problem 49 we can represent both linear transformations and translationsby matrix multiplication at the expense of using higher dimensions. This is done bythe homogeneous coordinates. I will illustrate in R* where most interest in this isfound. For each vector v = (vy .V2,V3)) consider the vector in R* (v1,v2,v3, 1)".What happens when you do1 0 0 a\ V{0 10a v2 |0 0 1 a3 V3 ,000 1 1Describe how to consider both linear transformations and translations all at once byforming appropriate 4 x 4 matrices.18.4 Subspaces Spans and BasesThe span of some vectors consists of all linear combinations of these vectors. As explainedearlier, a linear combination of vectors is just a finite sum of scalars times vectors.