The transpose of matrix (begin{bmatrix}1&0&-7 0&-2&3 4&5&6 end{bmatrix})
a.(begin{bmatrix}1&0&4 0&-2&5 -7&3&6 end{bmatrix})
b.(-83)
c.(-59)
d.(begin{bmatrix}-1&0&-4 0&2&-5 7&-3&-6 end{bmatrix})

If V = P3 with the inner product < f, g >=
R 1
11
f(x)g(x)dx, apply the Gram-Schmidt algorithm
to obtain an orthogonal basis from B = {1, x, x2
, x3}

Let V = M2×w(R) and W = P2(R). Define T
a b
c d
= a + b + (c − d)x + bx2
.
Let
β =
1 0
0 0
,
0 1
0 0
,
0 0
1 0
,
0 0
0 1
and
γ = {1, x, x2
}.
Find φβ and φγ. Find the matrix A so that LAφβ = φγT. Support your answer
by evaluating both maps on M =
3 1
−1 4

Suppose T : V → V is linear and let W be a subspace of V . Further suppose that
T(w) ∈ W for all w ∈ W. Let S : W → W be defined by S(w) = T(w).
(a) Find an example of such a T and {0} 6= W 6= V when V = R
2
.
(b) Prove (in general) that N(S) = N(T) ∩ W.

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.