(c) Find two different linear combinations of the vectors u...

(c) Find two different linear combinations of the vectors $u = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$, $v = \begin{pmatrix} 2 \\ 7 \end{pmatrix}$, $w = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$ that produce $b = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Now, if you take any three vectors $u, v, w \in \mathbb{R}^2$, will there always be two distinct linear combinations which produce $b = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

Transcript

00:01 Hi, in this question in the first one, in that the a1, we have 3u minus 2v plus 4w to be equal to minus 1 .2 .3.

00:14 So this implies v will be equal to 1 by 2 into 3u plus 4w minus of minus 1 .2 .3 and that will be equal to 1 by 2 into 3u plus 4w minus of minus 1 .2 .3 and that will be equal to 1.

00:30 1 by 2, 11 .3.

00:33 Minus 2.

00:34 In the b part, we have been given minus 2 u plus 4 v minus 5 w.

00:43 This will be equal to minus 4.

00:47 Minus 6 .2 plus 22 .6 minus 4 minus 4 minus 5 .5 .5 and this will be equal to.

01:00 That is 13 .5.

01:02 Minus 7 and your v is equal to 1 by 2 into 11 .3 comma minus 2.

01:08 And now for the second part the u vectors are given to us.

01:13 So we have shaded in all the linear combinations of c u plus d w for all the ab and c.

01:22 Moving ahead to the third question.

01:24 So for the third question in the a part, we'll say, that op vector will be equal to half of the og vector so that will be equal to 1 by 2 of this will be 1 0 0 plus 0 0 so that is equal to half i plus half j then in the b part we have o q vector that will be 1 by 2 2 od vector which is equal to 1 by 2 of ob b vector plus oc vector and this will be equal to 1 by 2 k plus 1 by 2 j then in the c part we have o r vector that is equal to 1 by 2 o e vector this will be equal to half of oa vector that is equal to 1 by 2 o e vector this will be equal to half of o a vector plus oc vector that will be equal to 1 by 2 i plus 1 by 2 k then further we'll say that in the d part we have os vector so that will be equal to ob vector plus 1 by 2 bf vector upon solving it this we'll be getting it as 1 by 2 i plus j plus 1 by 2 k for the e part we have o t vector that is equal to o c vector plus half c f vector so upon solving this is equal to 1 by 2 i plus 1 by 2 j plus k then in the f part we have to find oh u vector so this is equal to o a vector plus half a f vector when we solve it this is equal to i plus 1 by 2 j plus 1 by 2 k and therefore op is equal to 1 by 2 i plus 1 by 2 j b o q is equal to 1 by 2 j plus 1 by 2 k c o u u .s equal to 1 by 2 j plus 1 by 2 k c c o i is equal to 1 by 2 i plus 1 by 2 k d os is equal to 1 by 2 i plus j plus 1 by 2 k e o t is equal to 1 by 2 i plus 1 by 2 j plus 1 by 2 k now in the 4th problem we'll say that for the a o b d c this can be written as c j plus d k such that c is great than equal to 0 less than equal to 1 d greater than equal to 0 less than equal to 1.

04:38 Similarly, b will be oa, e, c, this will be equal to c i plus d k, this is such that c greater than equal to 0 less than equal to 1...

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