Vector Spaces (MA1101)
Tutorial Sheet 2
Q1.* Show that if v and w are linearly independent vectors in V , then so are v + w and
v − w.
Q2. Let S1 be a linearly dependent subset of a vector space V and S2 be such that
S2 ⊆ S1 . Then prove that S2 is linearly dependent. State and prove a similar property for
linear independence.
Q3. Let V := Rn . If x = (x1 , . . . , xn ) ∈ Rn , we call xj , the jth coordinate of x. Let
ei := (0, . . . , 0, 1, 0, . . . , 0) be the vector whose jth coordinate is zero unless j = i in which
case it is 1. Show that {ei | 1 ≤ i ≤ n} is a basis of V . This is called the standard basis of
Rn .
Q4.* Let V := M (n, R). Let Eij be the element of V whose (i, j)th entry is one and the
rest are zero. In notation, if Eij = (xrs ), we have
(
0, if r ̸= i and s ̸= j,
xrs =
1, if r = i and s = j.
Show that {Eij | 1 ≤ i, j ≤ n} is a basis for M (n, R).
Q5. Show that {(1, 2), (4, 3)} is a basis of R2 .
Q6. When is {v, w} a basis of R2 ?
Q7.* Show that {X, 3X 2 , 5 + X} is a basis of P2 . What about {2X, X 2 − 3X, 2X 2 }?
Q8.* Let Sn denote the set of symmetric matrices in M (n, R). Then Sn is a vector
subspace of M (n, R). What is its dimension?
(Hint: Let
a11 a12
∈ S2 .
A=
a21 a22
In general, to “locate” any matrix
x11 x12
X=
∈ M2×2 ,
x21 x22
we need four coordinates (x11 , x12 , x21 , x22 ). But, however, if A is symmetric, if we know a12
then we know a21 . Thus to “locate” A, we need only three coordinates a11 , a12 , a22 . Therefore
we expect that the dimension of S2 is three. To see this, recall the basis {Eij } of M (n, R)
where Eij is a matrix whose ijth entry is 1 and all the other entries are zero. If A ∈ Sn , then
A = a11 E11 + a12 E12 + a21 E21 + a22 E22 = a11 E11 + a12 (E12 + E21 ) + a22 E22 .
Thus {E11 , E12 + E21 , E22 } generates S2 . It is easy to see that it is linearly independent and
hence a basis.)
Q9. Find a basis for Sn of symmetric matrices.
Q10.* Let V = M (n, R) and let An be the set of all skew-symmetric matrices. Find a
basis and dimension of An .
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Q11. Let V = M (n, R)Pand W = {X ∈ V | trace X = 0}. Find a basis and dimension
of W . (Recall trace(X) = i xii if X = (xij ).)
Q12. Find a basis of C, considered as a vector space over R (see Exercise 2.1.21). Hint:
Any z ∈ C can be written as z = 1 · x + y · i, x, y ∈ R.
Q13. Can you exhibit a basis of Pn consisting of elements all of degree n? All of degree
≤ n − 1?
Q14.* Find the dimension of the set of solutions of
(1) x + 4z + t = 0, x + y + 2z − 4t = 0.
(2) x + 2y = 0, y − z = 0, x + y + z = 0.
Q15.* Find a basis for the vector space Mm×n (R) consisting of the set of m × n matrices
with real entries.
Q16. Show that the following elementary operations on a subset {v1 , . . . , vk } of a vector
space V “preserve” linear independence or dependence of the family:
(a) Interchanging two of the vectors.
(b) Multiplying a vector by a non-zero scalar.
(c) Replacing any vi by vi + αvj for any scalar α and any j ̸= i.
(Hint:What you are supposed to do is this: If we have {v1 , . . . , vi , . . . , vj , . . . , vn } and use
the first elementary operation to get {v1 , . . . , vj , . . . , vi , . . . , vn } then the first set is linearly
dependent (respectively independent) if and only if the second is so. Similar remarks apply
to others.)
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