Please solve these problems with full steps Let and Write y as the sum of a vect

Question

Please solve these problems with full steps

Let and Write y as the sum of a vector in Span{u} and a vector orthogonal to u. Let {v1,v2} be an orthogonal set of two nonzero vectors, and let c1 c2 be any nonzero scalars. Show that {c1v1, c2v2} is also an orthogonal set of vectors. (Note: You are showing that if a set of two nonzero vectors is orthogonal, then the set containing corresponding scalar multiples of those two vectors is also orthogonal. This result is true for orthogonal sets of any number of nonzero vectors.) Explain how to normalize a set of vectors. Explain why parts (a.) and (b.) show the following. "If the vectors in an orthogonal set are normalized, then the new set of vectors will be orthonormal." Show that an mTimesn matrix has orthonormal columns if and only if UTU = In. Suppose that U isan nTimesn matrix with orthonormal columns. Is U invertible? If so, what is U-1? Explain.

Explanation / Answer

Q1.

span of u will be some constant time u.

let that constant be a.

now let a vector orthogonal to u be the column vector [b

c]

then 3*b+c=0

-2=3*a+b

5=a+c

solving these 3 equations

we get

a=-0.1

b=-1.7

c=5.1

so Y=a*u+ [b

c]

Y= [-0.3 [-1.7

-0.1] + 5.1]

2)v1 and v2 are orthogonal.

then v1(dot)v2=0

consider (c1V1)(dot)(C2V2)=(c1*c2)*(V1(dot)V2)=0

so c1v1 and c2*v2 are also orthogonal.

b)to normalize,

first take the square root of sum of square of all the elements.

this is called the norm.

then divide each element by norm.

you will get the normalized vector.

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