Linear Algebra HomeWork
Vampass
linear_algerbra_review.pdf
July 05, 2017 MATH 262, Summer 2017 Emmanuel YOMBA
Review Sheet 2
Due Wed. July 5, 2017, in class.
1.) Find a basis for the solutions to the following system of linear equation
x1 + 2x2 −x3 + x4 = 0
−x1 − 2x2 + 3x3 + 5x4 = 0
−x1 − 2x2 −x3 − 7x4 = 0
2) Consider the following subspace of R4 :
S = Span
1 2 1 3
,
3 6 3 9
,
1 3 5 4
,
2 3 −2 5
Find a basis for S. 3.) Find a basis for the null space, row space and column space of A, if
A =
1 1 2 22 2 5 5
0 0 3 3
4.) It is given that A =
0 −3 −6 4 9 −1 −2 −1 3 1 −2 −3 0 3 −1 1 4 5 −9 −7
, rref(A) =
1 0 −3 0 5 0 1 2 0 −3 0 0 0 1 0 0 0 0 0 0
,
and rref(AT ) =
1 0 0 0 0 1 0 −5 0 0 1 2 0 0 0 0 0 0 0 0
a) Find the rank of A. b) Find a basis for the null space of A. c) Find a basis for the column space of A. We require that you choose the vectors for the basis from the column vectors of A.
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d) Find a basis for the row space of A. We require that you choose the vec- tors for the basis from the row vectors of A.
5.) A) Let B =
{[ 3 −5
] ,
[ 4 −6
]} , and B′ =
{[ 4 5
] ,
[ 6 7
]} .
Write down the matrices that take [~x]B′ to [~x]B and from [~x]B to [~x]B′ .
B) Suppose that B =
11
3
, 14
2
, 21
6
and B′ =
10
1
, 1−3
0
, 21
2
Find [~x]B, if [~x]B′ =
32
1
Answer [~x]B =
−12916
60
6.) a) Let S = {(x,y,z) ∈ R3 | 2x = 3z and y = −z} Is S a subspace of R3?
b) Consider the set of vectors S =
{[ x y
] : x + y ≥−5
} Is S a subspace of R2?
7.) Suppose A is a 9 × 4 matrix a) Is it possible for rank of A to equal 9? b) If dimension of Column of A is 2, what is the dimension of Nul(A)? c) Is it possible that Dim(Nul(A)) = 0? d) Is S a subspace of R3.
8.) Let ~u1, ~u2, ~u3, ~u4, ~u5 represent the columns of matrix
A =
1 2 3 −1 34 −6 5 4 3
3 0 6 −1 3
The matrix A can be reduced to R =
1 0 2 0
3 2
0 1 1 2
0 3 2
0 0 0 1 3 2
a) Which of the following sets of vectors are linearly independent? Justify your answer. i) {~u1, ~u2 ~u3} ii) {~u1, ~u2, ~u5}, iii) {~u2, ~u4}. b) Which of the sets from a) form a basis for R3? Justify. c) Find a basis for Nul(A).
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d) Find a basis for Col(A) from column of A. e) Find the rank and nullity of A. f) Express each column of A that is not in your basis as a linear combination of your basis vectors.
9.) Let ~X1 =
1 1 1 1
, ~X2 =
1 2 2 2
, ~X3 =
3 5 6 6
, ~X4 =
1 2 3 3
, ~Y =
2 3 4 4
.
a) Find the dimension of span{~X1, ~X2, ~X3, ~X4}. b) Verify whether ~Y ∈ span{~X1, ~X2, ~X3, ~X4} Solution For both question, row reduce [ ~X1, ~X2, ~X3, ~X4|Y ] and find that the dimension of the span is 3 and ~Y is there.
10.) Suppose that A = {~X1, ~X2, ~X3} is a basis of a three dimensional subspace of Rn. Let
~Y1 = ~X1 + ~X2, ~Y2 = ~X2 + ~X3, ~Y3 = ~X1 + ~X3.
It is known that B = {~Y1, ~Y2, ~Y3} is another basis of the same subspace. If
[ ~X]A =
32
1
, find [ ~X]B. Solution [ ~X]B =
30
1
A =
1 1 2 22 2 5 5
0 0 3 3
11.) Suppose that ν ⊂ R4 has basis A =
1 1 0 0
,
0 1 1 0
,
0 0 1 1
.
a) What are the coordinates [ ~X]A of vector ~X =
1 2 3 2
b) Which vector ~X in R4 has [ ~X]A =
12
3
Solution a) [ ~X]A =
11
2
, b) ~X =
1 3 5 3
12.) Find a basis of the space V of all matrices B that commute with
3
A =
[ 0 1 2 3
]
Solution basis of V is
[ 1 0 0 1
] ,
[ 0 1 2 3
] is a subspace of M2,2
13.) Show that S =
{[ a b c d
] ∈ M2,2|a + b + c + d = 0
} is a subspace of M2,2.
14.) a) Determine whether (8,−2,−7) is a linear combination of vector (1, 1, 1), (−2, 0, 1), and (−1, 3, 5). b) Determine whether the polynomial defined by q(x) = x2 + x + 2 is a linear combination of the polynomials defined by p1(x) = x
2 + 5 and p2(x) = x2 + 2x− 1 c) Show that the set {(1,−2, 3), (1, 0, 1), (0, 1,−2)} of three vectors, spans R3. Solution a) Many solutions example 2(1, 1, 1)+(−5)(−2, 0, 1)+0(−1, 3, 5), b) Exactly one linear combination 1
2 p1 +
1 2 p2. c) Show we have a unique solution
here. 15.) a) Show that {(1, 1, 0), (1, 0, 1), (0, 1, 1)} is a linearly independent subset of R3. b) Determine if the set of matrices{[
1 1 1 −1
] ,
[ 1 −1 0 0
] ,
[ 0 2 1 −1
]} is linearly independent.
c) Show that the polynomials p1,p2 and p3 defined by p1(x) = x 2 +
1, p2(x) = 2x 2 + x − 1, p3(x) = x2 + x form a basis for P2. Find the
coordinates of x2 −x + 4 in that basis. Solution this coordinate is [3,−1, 0]T .
16) Build an orthonormal basis B for R4 with the following set: S = {(2, 1, 0,−1), (1, 0, 2,−1), (0,−2, 1, 0), (1, 0, 0, 0)} Solution{(
2√ 6 , 1√
6 , 0,− 1√
6
) , ( 0,− 1
3 √ 2 , 4 3 √ 2 ,− 1
3 √ 2
) , (
2√ 21 ,− 4√
21 ,− 1√
21 , 0 ) , (
1√ 7 , 1 3 √ 7 , 2 3 √ 7 , √ 7 3
)} .
17) Consider the real vector space V of all the real continous functions de- fined on the interval [a,b]. Let f and g ∈ V . Define < f,g >=
∫ b a f(t)g(t)dt
(a,b ∈ R) a) Show that <,> defines an inner product on V . b) With a = −π and b = π, show that the S = {1, cos t, sin t} is an orthogo- nal set in V , c) From the set S, give an orthonormal set B.
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18) Recall that the inner product Space V from Problem 17) or real con- tinuous functions with a = −1 and b = 1 that is, with the inner product < f,g >=
∫ 1 −1 f(t)g(t)dt
a) Now, S = {1, t, t2, t3} is a linearly independent set in V . Use this set to build an orthogonal set S′. b) Find the angle between f(t) = t3 − 3t + 1 and g(t) = t + 1 c) Find the projection of the function 1
2 t onto t + 1.
Solution a) S′ = {1, t, t2 − 1/3, t3 − (3/5)t} b) θ = Arcos (√
365 146
) 19) Find the basis of the subspace Q of R3 formed by all the vectors that
are orthogonal to the vector ~v = (2, 3, 5). Solution B = {(−3, 2, 0), (−5, 0, 2)}.
20) In the plane V defined by the equation x1 + 3x2 − 2x3 = 0, consider the basis B = {~a1,~a2} = {(−3, 1, 0), (−1, 1, 1)} . a) Construct another basis B′ =
{ ~b1,~b2
} of V , such that neither ~b1 nor ~b2
has any negative components. b) Find the change of basis matrix P from B′ to B. c) Find the change of basis matrix from B to B′.
d) Write an equation relating the matrices [~a1 ~a2], [~b1 ~b2], and P = PB′−→B.
21) Show that the set S of all f in C2[a,b] such that f
′′ (x) + f(x) = 0
for all x in [a,b] is a subspace of C2[a,b].
22) Show that S = {A ∈ M2,2 a12 = −a21} is a subspace of M2,2.
23) Show that S = {(x, 1) x is a real number} is not a subspace of R2
24) Suppose that the vectors ~V1 = (−2, 1, 0, 0, 0), ~V2 = (−4, 0,−3,−2, 1) are a basis for nullspace of a 4X5 matrix A. Find a vector ~X such that ~X 6= ~0, ~X 6= ~V1, ~X 6= ~V2 and A ~X = ~0.
25) Define
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p1(t) = 1 + t 3, p2(t) = t + t
3, p3(t) = 2 + 3t + t 2.
a) Is the set B = {p1(t), p2(t), p3(t)} a basis for P3? Explain. b) Is the vector q(t) = 3 + 4t + t2 + t3 in the space spanned by the vectors in B? Explain.
26) Is the set S = {(7a−5b + c,a−b + c,c−a,b + c,a,b−a)|a,b,c ∈ R} a subspace of R6? Explain.
27) a)For what values of h will the columns of A =
1 −2 h6 3 5 −2 4 3
spans
R3? b) For what values of h will the columns of A be linearly independent?
28) Let R3 have the inner product < ~a,~b >= a1b1 + 2a2b2 + 3a3b3.
Use the Gram-Schmidt process to transform the basis {~v1,~v2,~v3} into an orthonormal basis. ~v1 = (1, 1, 1), ~v2 = (1, 1, 0), ~v3 = (1, 0, 0).
Ans. ~u1 = 1 6 (1, 1, 1), ~u2 =
1 6 (1, 1,−1), ~u3 = 1√6 (2,−1, 0).
29) Let ~y = (7, 6) and ~u = (4, 2). Write ~y as a sum of two orthogonal vectors, one in span(~u) and one orthogonal to ~u . Ans. ~y = (8, 4) + (−1, 2).
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