Question

Let W = {(x, y, z, w) ∈ R 4 | x − z = 0 and y + 2z = 0}

(a) Find a basis for W.

(b) Apply the Gram-Schmidt algorithm to find an orthogonal basis for the subspace (2) U = Span{(1, 0, 1, 0),(1, 1, 0, 0),(0, 1, 0, 1)}.

Answer #1

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

Consider the following subset:
W =(x, y, z) ∈ R^3; z = 2x - y from R^3.
Of the following statements, only one is true. Which?
(1) W is not a subspace of R^3
(2) W is a subspace of R^3
and {(1, 0, 2), (0, 1, −1)} is a base of W
(3) W is a subspace of R^3
and {(1, 0, 2), (1, 1, −3)} is a base of W
(4) W is a subspace of R^3
and...

The T: R 4 → R 4 , given by T (x, y, z, w) = (x + y, y, z, 2z +
1) is a linear transformation? Justify that.

Is the set of all x, y, z such x+ 3y + 2z = 0 a subspace of R^3
? If so find a basis for the space.

a) Apply the Gram–Schmidt process to find an orthogonal basis
for S.
S=span{[110−1],[1301],[4220]}
b) Find projSu.
S = subspace in Exercise 14; u=[1010]
c) Find an orthonormal basis for S.
S= subspace in Exercise 14.

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

Let W be the subspace of R4
spanned by the vectors
u1 = (−1, 0, 1, 0),
u2 = (0, 1, 1, 0), and
u3 = (0, 0, 1, 1).
Use the Gram-Schmidt process to transform the basis
{u1, u2,
u3} into an orthonormal basis.

Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4
(a)
Let w = (0, 6,
4, 1). Find ||w||.
(b)
Let W be the
subspace spanned by the vectors
u1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).
Use the Gram-Schmidt process to transform the basis
{u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas.

5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that
p(-1)=0.
(a) Prove that S is a subspace of the vector space of all
polynomials.
(b) Find a basis for S.
(c) What is the dimension of S?
6.
Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2
=(1,2,-6,1),
?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2
=(3,1,2,-2). Prove that V=W.

w ′′ − y + 2z = 3e-x
−2w ′ + 2y ′ + z = 0
2w ′ − 2y + z ′ + 2z ′′ = 0
w(0) = 1, w′ (0) = 1, y(0) = 2, z(0) = 2, z′ (0) = −2

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