Linear dependent thiab ywj siab kab: txhais, piv txwv

Hauv kev tshaj tawm no, peb yuav txiav txim siab seb qhov sib xyaw ua ke ntawm cov hlua yog dab tsi, linearly dependent thiab cov hlua ywj siab. Peb tseem yuav muab piv txwv rau kev nkag siab zoo ntawm cov khoom siv theoretical.

Defining ib Linear Combination of Strings

Kev sib xyaw ua ke (LK) lub sij hawm s₁nrog₂, …, s_n Matrix A hu ua ib qho kev nthuav qhia ntawm daim ntawv hauv qab no:

αs₁ + αs₂ + ... + αs_n

Yog tias tag nrho cov coefficient α_i yog sib npaug rau xoom, ces LC yog tsis tseem ceeb. Hauv lwm lo lus, qhov tsis tseem ceeb linear ua ke sib npaug ntawm xoom kab.

Piv txwv li: 0 · s₁ + 0 · s₂ + 0 · s₃

Raws li, yog tias tsawg kawg yog ib qho ntawm cov coefficients α_i tsis sib npaug rau xoom, ces LC yog tsis tseem ceeb.

Piv txwv li: 0 · s₁ + 2 · s₂ + 0 · s₃

Linearly dependent thiab ywj siab kab

Txoj hlua system yog linearly nyob (LZ) yog tias muaj cov kab sib txuas tsis tseem ceeb ntawm lawv, uas yog sib npaug rau cov kab xoom.

Li no nws ua raws li qhov tsis tseem ceeb LC tuaj yeem nyob rau qee kis sib npaug rau xoom txoj hlua.

Txoj hlua system yog linearly ywj siab (LNZ) yog tias tsuas yog LC tsis tseem ceeb yog sib npaug rau txoj hlua null.

Sau ntawv:

• Nyob rau hauv ib square matrix, kab kab system yog LZ tsuas yog hais tias tus txiav txim ntawm no matrix yog xoom (lub = 0).
• Nyob rau hauv ib square matrix, kab kab system yog LIS nkaus xwb yog tias tus txiav txim ntawm no matrix tsis sib npaug rau xoom (lub ≠ 0).

Piv txwv ntawm qhov teeb meem

Cia peb kawm seb txoj hlua system puas yog {s₁ = {3 4};s₂ = {9}} linearly nyob.

Kev txiav txim siab:

1. Ua ntej, cia peb ua LC.

α₁{3 4} + ib₂{9}.

2. Tam sim no cia saib seb yuav tsum muaj dab tsi α₁ и α₂yog li ntawd cov linear ua ke sib npaug ntawm cov hlua null.

α₁{3 4} + ib₂{9} = 12 0.

3. Cia peb ua ib qho kev sib npaug:

4. Faib thawj qhov sib npaug los ntawm peb, qhov thib ob los ntawm plaub:

5. Kev daws ntawm qhov system no yog ib qho α₁ и α₂, Nrog α₁ = -3a₂.

Piv txwv li, yog tias α₂ = 2ces α₁ = -6. Peb hloov cov txiaj ntsig no rau hauv qhov system ntawm kev sib npaug saum toj no thiab tau txais:

teb: yog li cov kab s₁ и s₂ linearly nyob.