List five vectors in Span {v1,v2} For each vector, show the weights on v1 and v2 used to generate the vector and list the three entries of the vector. Do not make a sketch.

Question

List five vectors in Span  {v1,v2}  For each vector, show the weights on  v1​  and  v2  used to generate the vector and list the three entries of the vector. Do not make a sketch.

Willie · Accepted Answer

Step1  Remember that Span{v1,v2}is the set of all linear combinations of the vectors  v1  and  v2.That means that it is the collection of all vectors that can be written as  av1+bv2  for some numbers a and b.The numbers a and b are called the weights of the vector relative to  {v1,v2}  Step2  So to get any vector in the span, we just pick any two numbers a and b form the linear combination with those weights.  Step 3  Vector1:Here is the vector in the span with weights 1 and 2.  1[71−6]+2[−530]=[71−6]+[−1060]=[−37−6]  Vector2:Weights:0 and -2.   0[71−6]−2[−530]=[10−60]  Vector3:Weights:0 and 0.  0[71−6]−2[−530]=[000]  Vector4:Weights:5 and 7.  5[71−6]+7[−530]=[026−30]  Vector5:Weights:-3 and 1  −3[71−6]+1[−530]=[−26018].  Result:  [−47−6],[10−60],[000],[026−30],[−26018]

\text{List five vectors in Span }\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\

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