Question

first use Gram-Schmidt on x1, x2 to create orthogonal vectors v1 and v2 with the same span as x1, x2. Now use the formula

p =((y, v1)/(v1, v1))v1 + ((y, v2)/(v2, v2))v2

to compute the projection of y onto that span. Of course, replace the inner product with the dot product when working with standard vectors

2)

Compute the projection of y = (1, 2, 2, 2, 1) onto span (x1, x2) where

x1 =(1, 1, 1, 1, 1) x2 =(4, 1, 0, 1, 4)

The inner product to use is the usual dot product. (This will compute a best-fitting function that is quadratic with no linear term, fitting to the data (−2, 1),(−1, 2),(0, 2),(1, 2),(2, 1).)

Answer #1

Which is required projection

first use Gram-Schmidt on x1, x2 to create orthogonal vectors v1
and v2 with the same span as x1, x2. Now use the formula
p =((y, v1)/(v1, v1))v1 + ((y, v2)/(v2, v2))v2
to compute the projection of y onto that span. Of course,
replace the inner product with the dot product when working with
standard vectors
1)
Compute the projection of y = (1, 2, 3) onto span
(x1, x2) where
x1 =(1, 1, 1) x2 =(1, 0, 1)
The inner...

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

3. a. Consider R^2 with the Euclidean inner product (i.e. dot
product). Let
v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v.
b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with
the Euclidean
inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7,
5).
C.Let V be an inner product space. Suppose u is orthogonal to
both v
and w. Prove that for any scalars c and d,...

Exercise 6. Consider the following vectors in R3 . v1 = (1, −1,
0) v2 = (3, 2, −1) v3 = (3, 5, −2 ) (a) Verify
that the general vector u = (x, y, z) can be written as a linear
combination of v1, v2, and v3. (Hint : The coefficients will be
expressed as functions of the entries x, y and z of u.) Note : This
shows that Span{v1, v2, v3} = R3 . (b) Can R3 be...

Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...

(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...

Use the inner product (u, v) =
2u1v1 +
u2v2 in
R2 and the Gram-Schmidt orthonormalization
process to transform {(?2, 1), (2, 5)} into an orthonormal basis.
(Use the vectors in the order in which they are given.)
u1 = ___________
u2 = ___________

Use the inner product <u,v>= 2u1v1 + u2v2 in R2 and the
Gram-Schmidt orthonormalization process to transform {(2, ?1), (2,
6)} into an orthonormal basis. (Use the vectors in the order in
which they are given.)
u1 =
u2 =

Which of the following vectors are unit vectors with respect to
the inner product: <(x1, x2, x3) , (y1, y2, y3)> = 2x1y1 =
2x3y3 in R3?
A. (1, 0, 0) B. (1, 0, 0)/sqrt(2) C. (1,
0, 1)/sqrt(2) D. (1, 1, 0)/2
Select from the following:
1. Only A
2. Only B and D
3. Only A and C
4. All of A, B, C and D
5. None of the above
Thank you!

1) Consider two vectors A=[20, 4, -6] and B=[8, -2, 6].
a) compute their dot product A.B
b) Compute the angle between the two vectors.
c)Find length and sign of component of A over B (mean Comp A
over B)and draw its diagram.
d) Compute Vector projection of B over A (means Proj B over A)
and draw corresponding diagram.
e) Compute Orthogonal projection of A onto B.

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