Y = [ 1 and u = [ 6
3 ] 2 ]
Write y as the sum of a vector in Span {u} and a vector orthogonal to u.
y = z = ___ + ___. (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)

Take as the vector in Span(u) and as the orthogonal vector to u

Step-by-step explanation:

Given vectors a,b we can calculate the projection of a onto b by the formula

. where is the dot product of vectors.

For this example, we will take y = [1,3] and u = [6,2].

Let b be the projection of y onto u. That is

Now, we want to find an orthogonal vector w such that b+w = y. From this equation, we can find w by taking w = y-b. Then

We must check that w and u are orthogonal. Note that if the vectors a,b are orthogonal, then the vectors a, kb are also orthogonal for every non-zero value of k. So, to check that a,kb are orthogonal, it suffices to check that a,b are orthogonal. We know that two vectors are orthogonal if and only if their dot product is zero. In our case, take the vector (6,2) and (-1,3). Note that

So, by the previous analysis, w and u are orthogonal.

answer: 193

explanation:

200-7 = 193