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Let v1and v2be two nonzero position vectors in 2that are not scalar multiples of each other. Explain why, given any vector win 2, there are scalars c1and c2such that w=c1v1+c2v2.

Short Answer

Expert verified

There exists the scalarsc1andc2such thatw=c1v1+c2v2.

Step by step solution

01

Step 1:Given information

.v1andv2be two nonzero position vectors

02

Step 2:Explaination

The objective is to explain given any vectorwin2, there are scalarsc1andc2such that

w=c1v1+c2v2

The vectorsv1andv2in2are not the scalar multiple of each other. Thus, the vectors are

independent in2.

Thus, the vectorsv1andv2in2behave as basis of2

The space2is spanned by the vectorsv1andv2.

For any non-zero vectorwin2the vectorwis the linear combination of the vectorsv1andv2.

Thus, there exists the scalarsc1andc2such thatw=c1v1+c2v2.

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