Question

2. Find a basis for R 4 that contains the vectors X = (1, 0, 0, 1) and Y = (1, 1, 0, 1).

Note: I'm not looking for the orthogonal basis, im looking for a basis that contains those 2 vectors.

Answer #1

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 = ...

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)}.

#2. For the matrix A = 1 2 1 2 3 7 4 7 9 find the
following. (a) The null space N (A) and a basis for N (A). (b) The
range space R(AT ) and a basis for R(AT )
. #3. Consider the vectors −→x = k − 6 2k 1 and −→y =
2k 3 4 . Find the number k such that the vectors...

(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 ⊥...

Find two unit vectors orthogonal to both
2, 4, 1
and
−1, 1, 0
.

Determine whether the following set of vectors is a basis for
M2(R): (−2 0 1 2) , ( 0 1 −1 1) , (−5 0 5 −4 ) , ( 3 −2 0 −1 ).

1)
Find a basis for the column space of A=
2 -4 0 2 1
-1 2 1 2 3
1 -2 1 4 4
2) Are the following sets vector subspaces of R3?
a) W = {(a,b,|a|) ∈ R3 | a,b ∈ R}
b) V = {(x,y,z) ∈ R3 | x+y+z =0}

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

1.
a) Find a value of x other than 0 such that the vectors <-3x,
2x> and <4, x> are perpendicular
b) find the domain of the vector function r (t) = <sin (t),
ln (t), 1 / (t-1)>
c) Determine if the sequence converges or diverges, if it
converges determines its limit
ln (2 + e ^ n) / 2020n
d) Find the point (a, b, c) where the line x = 1-t, y = t, z = 1...

a) Find a basis for W2 = {(x, y, z) ∈ R 3 : x + y + z = 0} (over
R). Justify your answer. What is the dimension of W2?
b) Find a basis for W4 = {(x, y, z) ∈ R 3 : x = z, y = 0} (over
R). Justify your answer. What is the dimension of W4?

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