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caleb3989
caleb3989
18.04.2020 • 
Mathematics

(1) Let {v1,v2,v3} be a set of vectors in Rn . If u is Span {v1,v2,v3}, show that 3u is in Span {v1,v2,v3}.

(Hint: since you are told that u is in the span {v1,v2,v3}, you can automatically say that there are scalars c1,c2 and c3 so that u = c1v1+c2v2+c3v3. Your goal now is to find a way to write 3u as a linear combination of {v1,v2,v3}

(2) Let u, v1,v2,v3 and v4 be vectors in the Rn . If u can be written as a linear combination of v1,v2, and v3 show that u can also be written as a linear combination of v1,v2,v3 and v4 .

(Hint: Look at the previous question and remember you can use 0 as a coefficient in your equation.)

(3) Let u, v, w1, w2, and w3 be vectors in the Rn. If u and v can both be written as linear combinations of w1, w2 and w3 show that u+v can also be written as a linear combination of w1, w2, and w3.

(hint:this is similar to the previous ones. Remember to choose different letters for your coefficients for u and v)

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