Find a basis for W^⊥W = span( < b m a t r i x

Transcript

00:01 In this question, we are asked to find an autotormal basis for w, where w is a span of the vectors negative 1 -1 -0 -0.

00:10 And negative -101.

00:12 Let's call the vectors by u1 and u2.

00:20 So we want to find the vectors v1 and v2 so that they're orthogonal and they are still, they're span still equals to w.

00:32 And we're going to take v1 to be equal to u1.

00:36 And to get v2 what we are going to do is so here is a picture this is a vector u1 and let's say this is a vector u2 now let's we now we just took v1 to be equal to u1 now let's make an orthogonal projection of the vector u2 onto the vector u1 which is also a v1 and consider the vector which is perpendicular to and the sum of the projection of u2 onto v1 and let's call this vector z and z equals to the vector u2 so we can write u2 as a sum of the projection of u2 onto v1 plus z and the vector z is going to be our vector v2 so this equals to the projection of u2 on v1 plus v2 and from this formula since we know u2 and we can find the projection of u2 onto v1 we can find v2 v2 equals to u2 so let me write it here v2 equals to u2 minus the projection of u2 onto v1 which is basically same as u2 minus the projection of u2 onto u1 because v1 equals to u1.

02:49 Let's find the projection first.

02:52 The projection of u2 onto u1 equals to the dot product of u2 and u1 divided by the dot product of u1 by itself and multiplied by the vector u1 now the dot to get the dot product of u1 and u2 we simply need to multiply the corresponding coordinates and add them up.

03:18 We are going to get negative 1 times negative 1 plus 1 times 0 plus 0 times 1 divided by u1 multiplied by u1...

Question

Find a basis for $W^\perp$\ $W = \text{span}\left( \begin{bmatrix} 1\\2\\3\\4 \end{bmatrix}, \begin{bmatrix} 5\\6\\7\\8 \end{bmatrix} \right)$

Question

Find a basis for $W^\perp$\ $W = \text{span}\left( \begin{bmatrix} 1\\2\\3\\4 \end{bmatrix}, \begin{bmatrix} 5\\6\\7\\8 \end{bmatrix} \right)$

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