Linear Algebra Answers

1. 1. Find a basis for the subspace of R Superscript 3 that is spanned by the vectors

v1 = (1, 0, 0), v2 = (1, 1, 0), v3 = (3, 1, 0), v4 = (0, -2, 0). Choose the correct letter.

A. v1 and v2 form a basis for span {v1, v2, v3, v4}.
B . v2 and v3 form a basis for span {v1, v2, v3, v4}.
C. v3 and v4 form a basis for span {v1, v2, v3, v4}.
D. v1 and v3 form a basis for span {v1, v2, v3, v4}.
E. v2 and v4 form a basis for span {v1, v2, v3, v4}.
F. v1 and v4 form a basis for span {v1, v2, v3, v4}.

1. Let Upper T Subscript Upper A Baseline colon Upper R Superscript 3 Baseline right-arrow Upper R Superscript 3 be multiplication by A and find the dimension of the subspace of Upper R Superscript 3 consisting of all vectors x for which Upper T Subscript Upper A Baseline left-parenthesis x right-parenthesis equals 0.

a. A= 1 1 0
1 0 3
1 0 3
The dimension is????

b. A = 1 4 0
1 4 0
1 4 0
The dimension is???

c. A = 3 0 0
-3 3 0
3 3 3
The dimension is????

1. Determine the dimension of and a basis for the solution space of the homogeneous linear system.

2x subscript 1 + x subscript 2 + 2x subscript 3 = 0
x subscript 1 + 6x subscript 3 = 0
x subscript 2 + x subscript 3 = 0

2. Find a basis for the given subspace of R Superscript 3, and state its dimension for the plane 2 x - 3 y + 5 z = 0.

3. Let v1 = (1,-1,3,-6) and v2 = (-3,4,-12,24).
Find standard basis vectors for R4 that can be added to the set {v1, v2} to produce a basis for R4.

A. v3 = (1,0,0,0) and v4 = (0,1,0,0)
B. v3 = (0,0,1,0) and v4 = (1,0,0,0)
C. v3 = (1,0,0,0) and v4 = (0,0,0,1)
D. v3 = (0,1,0,0) and v4 = (0,0,0,1)

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 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.