Question

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

Answer #1

There is a mistake. The question should read “Find the
orthogonal projection of **v onto u**. Then write
**v** as the sum of two orthogonal vectors, one in
span{u} and one orthogonal to u.

We have proj_{v} (u) = [(v.u)/(u.u)]u =
[(6-10+3)/(4+25+1)]u = -(1/30)u = -(1/30) [2,-5,-1] =
[-1/15,1/6,1/30]. This vector, being a scalar multiple of u is in
span{u}.

Let x = [-1/15,1/6,1/30]. Then v-x = [3,2,-3]- [-1/15,1/6,1/30]= [ 46/15,11/6,-91/30] = y (say).

Then v = x+y, where x = [-1/15,1/6,1/30] is in span{u} and y = [ 46/15,11/6,-91/30] is orthogonal to u.

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

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

2.
a. Given u = (9,7) and v = (2,3), find the projection of u onto
v. (ordered pair)
b. Find the area of the parllelogram that has the given vectors
u = j and v = 2j + k as adjacent sides.

Consider the following.
u =
−6, −4, −7
, v =
3, 5, 2
(a) Find the projection of u onto
v.
(b) Find the vector component of u orthogonal to
v.

Find the orthogonal projection of v=[−2,10,−16,−19] onto the
subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

Given vector ? = 2? + 3?, ? = −5? + ? + ?. Find the
followings.
a) The projection of u onto v
b) A vector that is orthogonal to both u and v

Let u = ⟨1,3⟩ and v = ⟨4,1⟩.
(a) Find an exact expression and a numerical approximation for
the angle between u and v. (b) Find both the projection of u onto v
and the vector component of u orthogonal to v.
(c) Sketch u, v, and the two vectors you found in part
(b).

Find the projection of u = −i
+ j + k onto v =
2i + j − 7k.

For parts ( a ) − ( c
) , let u = 〈 2 , 4 , − 1 〉 and v = 〈 4 , − 2 , 1 〉 .
( a ) Find a unit
vector which is orthogonal to both u and v .
( b ) Find the vector
projection of u onto v .
( c ) Find the scalar
projection of u onto v .
For parts ( a ) − (...

True or False
If A is the matrix of a projection onto a line L in R 2 and the
vector x in R 2 is not the zero vector, then the vector x − Ax is
perpendicular to the vector x.
If vectors u, v, x and y are vectors in R 7 such that u = 2v +
0x − 3y, then a basis for span(u, v, x, y) is {u, v, y}.

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