Question

Show that GL(2, R) acts transitively on R^2 − {0}. Hint: Let v, w ∈ R^2 be any

nonzero vectors and consider polar coordinates.

Answer #1

Hello

Please let me know if you need more explanation.

Note: this can also be done with the definition of transitive action i.e. to show that action has only a single orbit.

Pfa

Let V = R^3 and let W ⊂ V be defined by W = span{(1, 1, 1),(2,
1, 0)}. Show that W is a plane containing the origin, and find the
equation of W.

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ? → R^?
and ? : ? → R^? . Let ℎ : ? × ? → R ?+? be defined by ℎ(u, v) =
(?(u), ?(v)). If ? is continuous at x ∈ ? and ? is continuous at y
∈ ? , then show that ℎ is continuous at (x, y) ∈ ? × ? .
Hint: Note that for any vectors z...

A proof of the Triangle Inequality for vectors (let v and w be
vectors in R^n, then ||v+w||<= ||v||+||w||)
WITHOUT using the Cauchy-Schwarz Inequality.
Properties of the dot product are okay to use, as are any theorems
or definition from prior classes (Calc 3 and prior). This is for a
first course in Linear Algebra.
I keep rolling the boulder up the hill only to end up at
Cauchy-Schwarz again. Thanks for any help.

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

A2. Let v be a fixed vector in an inner product space V. Let W
be the subset of V consisting of all vectors in V that are
orthogonal to v. In set language, W = { w LaTeX: \in
∈V: <w, v> = 0}. Show that W is a subspace of V. Then,
if V = R3, v = (1, 1, 1), and the inner product is the usual dot
product, find a basis for W.

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,...

Let V be a vector subspace of R^n for some n?N. Show that if
k>dim(V) then the set of any k vectors in V is dependent.

In R^2, let u = (1,-1) and v = (1,2).
a) Show that (u,v) form a basis. Call it B.
b) If we call x the coordinates along the canonical basis and y
the coordinates along the ordered B basis, find the matrix A such
that y = Ax.

Let V be a vector space of dimension n > 0, show that
(a) Any set of n linearly independent vectors in V forms a
basis.
(b) Any set of n vectors that span V forms a basis.

Let v = (v1, · · · , vn), w = (w1, · · · , wn) ? R^n and let
<v, w> denote the inner product on R n given by <v, w>=
v1w1 + · · · + vnwn. Prove that for any linear transformation T :
R^n ? R, there exists a fixed vector v ? R^n such that T(w) =
<v, w>

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