Question

Define basis vectors. Do the three vectors ?? = (1,2,3), ?? = (3,1,2) and ?? = (4,2,1) form a basis?

Answer #1

Find a subset of the given vectors that form a basis for the
space spanned by the vectors. Verify that the vectors you chose
form a basis by showing linear independence and span: v1
(1,3,-2), v2 (2,1,4), v3(3,-6,18),
v4(0,1,-1), v5(-2,1-,-6)

why
is it that when findinf the basis for a set of vectors you row
reduce the augmented matrix with zero and then find the pivot
columns ans the set of correspondingvectors in the original matrix
form the basis, however whej youre finding a basis for the
eigenspace, and you do A-lambda*I, why dont you do the same
procedure with that matrix?

Two planes are passing through point P(1,2,3). The normal
vectors to the planes are ? =〈4,1,2〉 and ? =〈2, 0, 0〉.
Find the equations of the planes
Find the equation of the line which is the intersection of the
planes quiz.

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 ⊥...

Define solid angle. How do you define one steradian on the basis
of the surface area of a sphere that subtends a given solid
angle.

Let U be the span of (1,2,3) in R^3. Describe a basis for the
quotient space R^3 / U.

What does it mean if a set of basis vectors is complete?
a. The only vector that is orthogonal to every basis vector is
the 0 vector
b. The inner product of any two basis vectors is 0
I was thinking it was B but how would it be justified

3. (a) Consider R 3 over R. Show that the vectors (1,
2, 3) and (3, 2, 1) are linearly independent. Explain why they do
not form a basis for R 3 .
(b) Consider R 2 over R. Show that the vectors (1, 2),
(1, 3) and (1, 4) span R 2 . Explain why they do not form a basis
for R 2 .

Let the set of vectors {v1, ...vk, ...
,vn} are basis for subspace V in Rn.
Are the vectors v1 , .... , vk are
linearly independent too?

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