List five vectors in Span {v1,v2}. For each vector, show the weights on v1 and v2 used to generate the vector and list three entries of the vector. Do not make a seketch. v1=[71−6], v2=[−530]

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 three entries of the vector. Do not make a seketch. v1=[71−6], v2=[−530]

diskusje5 · Accepted Answer

Remember that span {v1,v2} is the set of all linear combinitions 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 So to get any vector in the span, we just pick any two numbers a and b and from the linear combination with those weights. Vector 1: Here is the vector in the span with weights 1 and 2. The three entries of the vector are given in red: 1[71−6]+2[−530]=[71−6]+[−1060]=[−37−6] Vector 2: Weights: 0 and -2. The three entries of the vector are given in red: 0[71−6]−2[−530]=[10−60] Vector 3: Weight 0 and 0. The three entries of the vector are given in red: 0[71−6]+0[−530]=[000] Vector 4: Weights 5 and 7. The three entries of the vector are given in red: 5[71−6]+7[−530]=[026−30] Vector 4: Weights -3 and 1. The three entries of the vector are given in red: −3[71−6]+1[−530]=[−26018] Result: [−37−6],[10−60],[000],[026−30],[−26018]

Mr Solver · Accepted Answer

Answer:Vector 1: [−111−12]Vector 2: [−7−16]Vector 3: [−530]Vector 4: [38−2−24]Vector 5: [−31318]Explanation:Let&#039;s start by expressing the vectors in the span using the weights c1 and c2:Vector 1: c1·v1+c2·v2=c1·[71−6]+c2·[−530]Vector 2: c1·v1+c2·v2=c1·[71−6]+c2·[−530]Vector 3: c1·v1+c2·v2=c1·[71−6]+c2·[−530]Vector 4: c1·v1+c2·v2=c1·[71−6]+c2·[−530]Vector 5: c1·v1+c2·v2=c1·[71−6]+c2·[−530]Now, let&#039;s choose specific values for c1 and c2 to obtain the vectors:For Vector 1, let&#039;s use c1=2 and c2=3:Vector 1: 2·[71−6]+3·[−530]=[142−12]+[−1590]=[−111−12]For Vector 2, let&#039;s use c1=−1 and c2=0:Vector 2: −1·[71−6]+0·[−530]=[−7−16]+[000]=[−7−16]For Vector 3, let&#039;s use c1=0 and c2=1:Vector 3: 0·[71−6]+1·[−530]=[000]+[−530]=[−530]For Vector 4, let&#039;s use c1=4 and c2=−2:Vector 4: 4·[71−6]−2·[−530]=[284−24]−[−1060]=[38−2−24]For Vector 5, let&#039;s use c1=−3 and c2=2:Vector 5: −3·[71−6]+2·[−530]=[−21−318]+[−1060]=[−31318]

Eliza Beth13 · Accepted Answer

To solve the problem, we need to find five vectors in the span of v1 and v2. The span of v1 and v2 is the set of all possible linear combinations of these two vectors.Let&#039;s denote the weights on v1 and v2 as a and b respectively. We can generate a vector in the span as follows:Vector&amp;nbsp;1:&amp;nbsp;a=0,b=0⇒𝐯1=[000]Vector&amp;nbsp;2:&amp;nbsp;a=1,b=0⇒𝐯2=[71−6]Vector&amp;nbsp;3:&amp;nbsp;a=0,b=1⇒𝐯3=[−530]Vector&amp;nbsp;4:&amp;nbsp;a=2,b=−3⇒𝐯4=[19−7−12]Vector&amp;nbsp;5:&amp;nbsp;a=−4,b=2⇒𝐯5=[−33−512]These are five vectors in the span of {v1,v2}, along with their respective weights on v1 and v2 and three entries of each vector.

madeleinejames20 · Accepted Answer

Step 1:Given:v1=[71−6] and v2=[−530].To find vectors in the span of v₁ and v₂, we can express them as linear combinations of v₁ and v₂ using weights or scalar multipliers. Let&#039;s denote the weights on v₁ and v₂ as a and b, respectively.The general form of a vector in the span of v₁ and v₂ can be written as:v=a·v1+b·v2.Step 2:Now, we can find five vectors in the span:1. Vector 1: Let&#039;s choose a=0 and b=0, then:v1=0·[71−6]+0·[−530]=[000].Three entries of v1 are 0, 0, and 0.2. Vector 2: Let&#039;s choose a=1 and b=0, then:v2=1·[71−6]+0·[−530]=[71−6].Three entries of v2 are 7, 1, and −6.3. Vector 3: Let&#039;s choose a=0 and b=1, then:v3=0·[71−6]+1·[−530]=[−530].Three entries of v3 are −5, 3, and 0.4. Vector 4: Let&#039;s choose a=2 and b=1, then:v4=2·[71−6]+1·[−530]=[95−12].Three entries of v4 are 9, 5, and −12.5. Vector 5: Let&#039;s choose a=−3 and b=2, then:v5=−3·[71−6]+2·[−530]=[−29−112].Three entries of v5 are −29, −1, and 12.Step 3:Therefore, the five vectors in the span of {v1,v2}, along with the weights on v1 and v2 used to generate each vector and three entries of each vector, are:v1=[000] (weights: 0 and 0), entries: 0, 0, and 0.v2=[71−6] (weights: 1 and 0), entries: 7, 1, and −6.v3=[−530] (weights: 0 and 1), entries: −5, 3, and 0.v4=[95−12] (weights: 2 and 1), entries: 9, 5, and −12.v5=[−29−112] (weights: −3 and 2), entries: −29, −1, and 12.

List five vectors in Span \{v_1,v_2\}. For each vector, show the weights o

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