Linear Algebra Answers

Find the orthogonal canonical reduction of

the quadratic form Q = 3x2+ 2y2— 2.5 xy.

Also give its principal axes. Finally, draw a

rough sketch of the orthogonal canonical

reduction of Q = 4.

5. Explain why the set S is not a basis for R3 .


u1= (-1, 3, 2), u2= ( 6,1,1) for R^3.



6. Given matrix A in row cchelon form (R),

find the (a) basis for row space of A and rank (A).

(b) basis for the solution space of A and nullity (A).


1 -2 0 0 3

R= 0 1 3 2 0

0 0 1 1 0

0 0 0 0 0

1. Determine whether the set of all vectors of the form (a, b ,c) such that b=a+c+1 is a subspace of R3 .


2. Determine whether the vector p is in the span {S}. Given S = { p_1,p_2,p_3} where

p_1=2+x+〖4x〗^2, p_2=1-x+〖3x〗^2, p_3=3+2x+〖5x〗^2; p=7+8x+9x^2.


3. Using two methods, verify that the set S is linearly independent.

v1 = (2, -2, 0), v2=(6,1,4), v3=(2,0,-4)



4. Express v as a linear combination of the vectors v_(1,), v_2,v_3.


v=(2,-1,3), v1=(1,0,0), v2=(2,2,0), v3=(3,3,3)

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 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.