Let Bold u comma Bold v comma Bold w 1 comma Bold w 2 comma and Bold w 3 be vectors in Bold Upper R Superscript n. If u and v can both be wr

Question

Let Bold u comma Bold v comma Bold w 1 comma Bold w 2 comma and Bold w 3 be vectors in Bold Upper R Superscript n. If u and v can both be written as linear combinations of  Bold w 1 comma Bold w 2 comma and Bold w 3​, show that u​+v can also be written as a linear combination of Bold w 1 comma Bold w 2 comma and Bold w 3.

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Mary 3 weeks 2022-01-03T06:46:25+00:00 1 Answer 0 views 0

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    2022-01-03T06:47:56+00:00

    Answer:

    Step-by-step explanation:

    given that U, V are two vectors in R^n

    These two vectors can be written as a linear combination of 3 vectors

    w1, w2, and w3

    To prove that  U+V also can be written as a linear combination of these three vectors.

    Since U is a linear combination we can write for not all a,b, c equal to 0

    U = aw1+bw2+cw3

    Similarly for d,e,f not all equal to 0

    V= cw1+dw2+ew3

    Adding these we have

    U+V =(a+d)w1 + (b+e) w2+(c+f)w3

    Here all a+d, b+e or c+f cannot be simultaneously 0.

    So we get U+V can be written as a linear combination of w1, w2 w3 as follows:

    U+W = gw1+hw2+iw3 \\g = a+d\\h = b+e\\i = c+f

    Proved

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45:7+7-4:2-5:5*4+35:2 =? ( )