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.

Answer 1

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