Let v = ? 1 ?, and v1 = ? 1 ?, v2 = ? 1 ?, v3 = ? 5 ?. ...

Let v = ? 1 ?, and v1 = ? 1 ?, v2 = ? 1 ?, v3 = ? 5 ?. | 1 | | 2 | |-1 | |-2 | ?-1 ? ?-1 ? ?-1 ? ? 1 ? Given the fact that B = (v1, v2, v3) is a basis for R^3, (1) Use the Gram-Schmidt process to convert B into an orthogonal basis B' = (u1, u2, u3) for R^3. (2) Transform B' = (u1, u2, u3) into an orthonormal basis B'' = (w1, w2, w3) for R^3. (3) Find [v]B', the coordinate vector of v relative to the basis B'.

Transcript

00:01 Hi, in this question, it is given that vector v is equal to 1 1 minus 1 and v1 is equal to 1 2 minus 1 and v2 is equal to 1 minus 1 minus 1 and vector v3 is equal to 5 minus 2 1 and in the first part of the question, we need to use gram -schmidt rule to convert this basis function which is a function of v1 v2 v3 into b dash which is orthogonal basis.

01:00 So, using this gram -schmidt rule, w1 is equal to v1 which is equal to 1 2 minus 1 and then w2 is equal to v2 minus dot product of w1 and v2 divided by magnitude of w1 square multiplied by w1 and w3 is equal to v3 minus dot product of w1 and v3 divided by magnitude of w1 square multiplied by w1 minus w2 v3 dot product divided by magnitude of w2 square multiplied by w2.

02:09 So, here the dot product of w1 and v2 is equal to 1 2 minus 1 and 1 minus 1 and minus 1.

02:24 So, this will be 1 minus 2 plus 1.

02:27 So, this is equal to 0 and w2 is equal to v2 which is 1 minus 1 and minus 1.

02:37 So, magnitude of w2 is equal to square root of 1 square minus 1 square plus minus 1 square.

02:45 So, this will be 1 plus 1 plus 1 which is square root of 3.

02:50 Now, dot product of w1 v3 is equal to 1 2 minus 1 and then 5 minus 2 1.

03:03 So, this will be equal to 5 minus 4 minus 1.

03:07 So, this is also equal to 0 and then w2 v3 dot product is 1 minus 1 minus 1 dot product with 5 minus 2 1.

03:23 So, this is 5 plus 2 minus 1 which is equal to 6.

03:29 So, w3 will be equal to 5 minus 2 1 minus 0 minus 6 divided by 3 of 1 minus 1 minus 1.

03:44 So, this is equal to 3 0 3...

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