Question

Prove that the orthogonal projection on the span of vectors that are not orthogonal can be reduced to solving normal equations. Please give an example whatever you like.

Answer #1

Suppose is orthogonal to vectors and . Show that is orthogonal
to every in the span {u,v}.
[Hint: An Arbitrary w in Span
{u,v} has the
form
w=c1u+c2v.
Show that y is orthogonal to such a
vector w.]
What theorem can I use for this?

Prove that an orthogonal projection is a positive operator.

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

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

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...

Vectors u1= [1,1,1] and u2=[8,-7,-1] are
perpendicular. Find the orthogonal projection of
u3=[65,-19,-31] onto the plane spanned by u1
and u2.

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,...

Prove that the span of three linearly independent vectors, u, v,
w is R3

Please describe in one or two sentences what an orthogonal
projection is.
Also, do the same for residual projection.
Thank you!

Linear Algebra
What does it mean for vectors to span Rm? Please give
a precise and complete definition. Thank you!

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