# Solved: the test cross ab/ab x ab/ab is performed

I am currently studying Halmos" "Linear Algebra Problem Book" & am stuchồng on problem 21(4).

Let \$V\$ be the phối \$acsantangelo1907.combbR_+\$, & let \$F\$ be the phối \$acsantangelo1907.combbR\$.

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Let"s define the sum of two positive numbers: \$\$a+b = ab,\$\$and the product of a positive number \$a\$ and a real number \$b\$: \$\$a*b = b^a.\$\$Prove sầu that \$V\$ is a vector-space.

Proving that addition is commutative sầu, associative, as well as the existence of an additive sầu identity (which is equal khổng lồ \$1\$) and an inverse (which is equal lớn \$1/x\$) wasn"t problematic.

However, multiplication & distributivity caused a few problems. Is the multiplicative identity also \$1\$, since \$x^1 = x\$?

Is \$(ab) x = x^b^a\$, in which case isn"t \$a (bx) = x^b^a = x^ba\$?

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edited Jul 15 "15 at 11:40
M Grande
asked Jul 15 "15 at 0:42

M GrandeM Grande
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Yes, the scalar \$1 in Bbb R\$ is the multiplicative identity for the vector space, because \$x^1=x\$, as you wrote.

Actually, \$(ab)x eq x^a^b\$ because since \$a,b\$ are scalars, their product is the usual multiplication on \$Bbb R\$, not the scalar multiplication as defined for a scalar & a vector.

Hence, \$(ab)x=x^ab=x^ba=(x^b)^a=a(x^b)=a(bx)\$, so scalar multiplication is associative sầu.

For distributivity, what you need is \$(a+b)x=ax+bx\$ & \$a(x+y)=ax+yx\$.

Xem thêm: Bài Văn Tả Cây Che Bóng Mát Lớp 4,5, Ngắn Gọn, Tả Cây Có Bóng Mát Lớp 5

The first one: \$(a+b)x=x^a+b=x^ax^b=ax+bx\$

The second one: \$a(x+y)=a(xy)=(xy)^a=x^ay^a=ax+ay\$.

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answered Jul 15 "15 at 0:59

coldnumbercoldnumber
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Distributivity is just \$(ab)^c=a^cb^c\$. Associativity (of the action) is \$(a^b)^c=a^bc\$.

\$(ab)x=x^ab\$, by the definition of the action. \$a(b(x))\$ is \$(x^b)^a=x^ba\$, which you should be careful khổng lồ distinguish from \$x^b^a\$, since exponentiation is not associative.

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answered Jul 15 "15 at 0:58

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Different approach: note that\$\$a+_V b = exp(log(a) + log(b))\a*_V b = exp(bcdot log(a))\$\$

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answered Jul 15 "15 at 1:52

Ben GrossmannBen Grossmann
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Thanks for contributing an answer khổng lồ acsantangelo1907.comematics Stachồng Exchange!

But avoid

Asking for help, clarification, or responding khổng lồ other answers.Making statements based on opinion; baông chồng them up with references or personal experience.

Use acsantangelo1907.comJax to lớn format equations. acsantangelo1907.comJax reference.

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