i. In the following, let (3,-2-1).,-(1,1,-1), ald = K-3, 2.6). Assume always ref

Question

i. In the following, let (3,-2-1).,-(1,1,-1), ald = K-3, 2.6). Assume always refers to the dot product. Find the follow- ing, if possible; if not possible, explain why not. (a) 371-272+4 (b). (c) 271 7+27s (d)·· (e)1 . (172-273) (f) I (8) 1571 2. Let V = { 1,2. . . . . t, } bc a basis for a vector space V (a) Prove: V' will also be a basis for V, where we have replaced with t., a linear combination of the vectors t, 72. 7, with ak0, (This means you can replace any basis vector with any linear combination of that vector with the "preceding" basis vectors) (b) Let t = rit, +12 2, where we want thi to be orthogonal to 71. Find a possibility for 721, (c) Find a suitable replacement for the second vector in so that the revised sct of vectors is orthogonal. (d) Find a suitable replacement for the second and third vectors of so that the revised set of vectors are orthogonal. Note: Feel free to use the replacement vector you found in the previous question. If you trust it.

Explanation / Answer

1)

3v1 = ( 9 , -6, -3)

-2v2 = (-2,-2,2)

4v3 = (-12,8,24)

3v1 - 2v2 + 4v3

= (9,-6,-3) + (-2,2,2) + (-12,8,24)

= (-5 , 4, 23)

2)

v1.v2 = 3* 1 + -2* 1 + -1 * -1

= 3 -2 +1

= 2

3)

not possible since 2 v1.v2 is a scalar and 2v3 is vector and once cannot add a scalar to vector

d)

not possible , since v1.v2 is scalar and v3 is vector

e)

3v2 = (3,3,-3)

-2v3 = ( 6, -4, -12)

3v2-2v3 = ( 9,-1,-15 )

v1.(3v2 - 2v3)

= (3,-2,-1) . (9,-1,-15)

= 3*9 + -2 * -1 + -1 * -15

= 27 + 2 + 15

= 44

f)

|v1| = sqrt( 3^2 + (-2)^2 + (-1)^2)

= sqrt(14)

g)

5v1 = (15,-10,-5)

|5v1| = sqrt( 15^2 + 10^2 + 5^2)

= sqrt(350)

= 5 sqrt(14)

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