Linear Algebra Answers

Let V be the vector space of all sequences over R. Given (a1, a2, . . .) ∈ V , define
T, U : V → V by
T(a1, a2, a3, a4, . . .) = (a1, a3, a5, . . .) and U(a1, a2, a3, a4, . . .) = (0, a1, 0, a2, 0, a3, . . .)
(a) Find N(T) and N(U).
(b) Explain why T is onto, but not 1-1.
(c) Explain why U is 1-1, but not onto.

Let L = {(1, 1, 1, 1, −4),(1, −1, 3, −2, −1)}. Find 6 vectors in the collection,
say H, such that L ∪ H spans the entire space.

Check p(x) + p(−x) ∈ P
(e)
for every p(x) ∈ R(x). Check that the map
ψ : R[x] → P
(e) given by ψ(p(x)) = p(x)+p(−x)
2
is a linear map. Further, check that
ψ
2 = ψ. Determine the kernel of ψ.

given that\\(A=\\begin{pmatrix}1 & 2 & 3\\\\ 4 & 5 & 6 \\end{pmatrix}\\)\nand \\[B=\\begin{pmatrix} 1 & 2\\\\ 3& 4\\\\ 5& 6 \\end{pmatrix}\\]\n. Find AB

Determine if the matrix p=
[√3/3, √6/6, -√2/2 ]
[-√3/3, √6/3, 0 ]
[√3/3, √6/6, √2/2] is othognal.

Determine if the matrix
q= √3/3, √6/6, -√2/2
-√3/3, √6/3, 0
√3/3, √6/6, √2/2
Is othognal.

Determine if the matrix p=
D= 1/√2, 0, 1/√2
0, 1, 0
-1/√2, 2, 1/√2
Is othognal.

Show that no skew symmetric matrix can be of rank 1

Would you able to do an assignment for tomorrow?

Q. Find the dimension of the subspace of R4 that is span of the vectors
(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))