Question #266494

Consider the basis for , where , , and , and let t:R^3 -> R^2 be the linear transformation for which T(v1)=(-1,0), T(v2)=(-1,1), T(v3)=(0,1). Find a formula for T(x1, x2, x3), and use that formula to find T(7,13,7)


1
Expert's answer
2021-11-16T11:21:16-0500

TT - linear transformation:

T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)

T(ku)=kT(u)T(ku)=kT(u)


1.

v=(x1,x2)R2v=(x_1,x_2) \in \mathbb{R} ^2 - arbitrary vector.

It can be expressed as a linear combination of v1v_1 and v2v_2:

v=av1+bv2v=av_1+bv_2

(x1,x2)=a(2,1)+b(1,3)=(2a+b,a+3b)(x_1,x_2)=a(-2,1)+b(1,3)=(-2a+b,a+3b)

{x1=2a+bx2=a+3b    {x1+2x2=7bx2=a+3b{b=x17+2x27a=3x17+x27\begin{cases} x_1=-2a+b \\ x_2=a+3b \end{cases} \; \; \begin{cases} x_1+2x_2=7b\\ x_2=a+3b \end{cases} \begin{cases} b=\frac{x_1}{7}+\frac{2x_2}{7} \\a=\frac{-3x_1}{7}+\frac{x_2}{7} \end{cases}


T(v)=T(av1+bv2)=aT(v1)+bT(v2)=a(1,2,0)+b(0,3,5)=T(v)=T(av_1+bv_2)=aT(v_1)+bT(v_2)=a(-1,2,0)+b(0,-3,5)=

=(3x17+x27)(1,2,0)+(x17+2x27)(0,3,5)==(\frac{-3x_1}{7}+\frac{x_2}{7})(-1,2,0)+(\frac{x_1}{7}+\frac{2x_2}{7} )(0,-3,5)=

=(3x17+x27,9x17+4x27,5x17+10x27)=17(3x1x2,9x14x2,5x1+10x2)=(\frac{3x_1}{7}+\frac{-x_2}{7} , \frac{-9x_1}{7}+\frac{-4x_2}{7} , \frac{5x_1}{7}+\frac{10x_2}{7})= \frac{1}{7}(3x_1-x_2,-9x_1-4x_2,5x_1+10x_2)


T(2,3)=17(9,6,20)=(97,67,207)T(2,-3)=\frac{1}{7}(9,-6,-20)=(\frac{9}7,-\frac{6}7,-\frac{20}7)


2.

v=(x1,x2,x3)R3v=(x_1,x_2,x_3)\in \mathbb{R}^3 - arbitrary vector

It can bee expressed as a linear combination of v1,v2,v3:v_1,v_2,v_3:

v=av1+bv2+cv3v=av_1+bv_2+cv_3

(x1,x2,x3)=a(1,1,1)+b(1,1,0)+c(1,0,0)=(a+b+c,a+b,a)(x_1,x_2,x_3)=a(1,1,1)+b(1,1,0)+c(1,0,0)=(a+b+c,a+b,a)

{x1=a+b+cx2=a+bx3=a{a=x3b=x2x3c=x1x2\begin{cases} x_1=a+b+c \\ x_2=a+b \\ x_3=a \end{cases} \begin{cases} a=x_3 \\ b=x_2-x_3 \\ c= x_1-x_2 \end{cases}


T(v)=T(av1+bv2+cv3)=aT(v1)+bT(v2)+cT(v3)=T(v)=T(av_1+bv_2+cv_3)=aT(v_1) +bT(v_2)+cT(v_3)=

=a(3,1,6)+b(4,0,1)+c(1,7,1)==a(3,-1,6)+b(4,0,1)+c(-1,7,1)=

=x3(3,1,6)+(x2x3)(4,0,1)+(x1x2)(1,7,1)== x_3(3,-1,6)+(x_2-x_3)(4,0,1)+(x_1-x_2)(-1,7,1)=

=(x1+5x2x3,7x17x2x3,x1+5x3)=(-x_1+5x_2-x_3,7x_1-7x_2-x_3, x_1+5x_3)

T(3,6,1)=(28,20,2)T(3,6,-1)=(28,-20,-2)


3.

T(4v15v2+6v3)=4T(v1)5T(v2)+6T(v3)=T(4v_1-5v_2+6v_3)=4T(v_1)-5T(v_2)+6T(v_3)=

=4(1,1,2)5(0,3,2)+6(3,1,2)=(14,13,10)=4(1,-1,2)-5(0,3,2)+6(-3,1,2)=(-14,-13,10)

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