PLease help? In Lecture 28, we derived the steps of the Gram-Schmidt algorithm.

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PLease help?

In Lecture 28, we derived the steps of the Gram-Schmidt algorithm. On slide 12, at the end of the page, it reads: "So {g1, g2} is an orthonormal basis". Show that this statement is true, i,e. show that ||g1|| =i,e. normality. ||g2|| =i,e. normality. = 0 i,e. orthogonality An orthonormal basis is a very good thing! Let's make one from a pile of vectors a friend gives us. Objective: Take the set of our friends basis vectors:{f1, ,fN} and create a set of good vectors: {g1, gN} that form an orthobasis. To put it another way, we want to have a result where Span {gk} = span {fk} =1 And = To put another way, we want to have a result where Span {gk} =span {f k} And = Let's take the second vector from our friend and add it in. Think about where we want g2 to be . . . The projection of f2 onto g1 is g1 We see that f2 = g1 +e2 So e2 =f2- g1 And we normalize e2 to give g2 =e2/||e2|| So {g1,g2} is an orthonormal basis for span {f1,f2} To put it another way, we want to have a result where Span {g k} = span {fk} And = 1 k=l 0 k l Let's take the first vector from our friend and say: e1 =f1 And let's normalize e1 i,e.g1 = e1/||e1|| So we now have g1 as an orthonormal Basis for span {f1}.

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