We will cover following topics

Linear Dependence and Independence
Vector Spaces and Subspaces
- Vector Space
- Vector Subspace
Basis and Dimension
- Basis
- Dimension
PYQs

Linear Dependence and Independence

We need to understand the concept of linear dependence and independence as the number of dependent vectors present no additional information and thus turn out to be redundant. In this section, we will study how to prove that a set of vectors is linearly dependent or independent.

A set of vectors $V = {v_{1}, v_{2}, . . ., v_{n}}$ is said to be linearly independent if $\sum_{i \in {1, 2,...,n}} a_{i} v_{i} = 0$ implies that $a_{i} = 0 \forall i \in {1,2,...n}$ . This implies that there is no vector in the set $V$ which can be expressed as a linear combination of other vectors in the set.

The set of vectors is otherwise called linearly dependent.

To check if a given set of vectors is linearly independent, reduce the row matrix of the given vectors to Echelon form. If there are no zero row vectors in the Echelon form, the set of vectors is said to be linearly independent, and if there are any zero row vectors in the set, then it is called linearly dependent.

Example 1: Prove that any set of vectors which contains the zero vector is linearly dependent.
Sol 1: Let $0$ be the zero vector, and $v_{1}$ , $v_{2}$ …, $v_{k}$ are the other vectors in the set.
Now, we have the non-trivial linear combination 1 $0$ + 0 $v_{1}$ + 0 $v_{2}$ + … + 0 $v_{k}$ . This is a non-trivial linear combination because one of the coefficients (the first one) is non-zero.

Thus by definition, the set ${0, v_{1}, . . ., v_{k}}$ is linearly dependent.

Vector Spaces and Subspaces

A vector space is simply a collection of vectors which can be added together and multiplied by a scalars. The resultant vector thus obtained also form part of the original vector space.

Vector subspaces are subsets of vector spaces.

Vector Space

Let $V$ be a non-empty set with two operations- vector addition and scalar multiplication. Then, $V$ is called a vector space if the following axioms hold true:

$(u + v) + w = (u + v) + w \forall u$ , $v$ , $w \in V$
$0 \in V$ such that $u + 0 = 0 + u = u$
For each $u \in V$ , there exists $(- u) \in V$ such that $u + (- u) = (- u) + u = 0$
For $u$ and $v \in V$ , $u + v = v + u$
$c (u + v) = c u + c v$ , for any scalar $c \in K$
$(a + b) u = a u + b u$ for any scalars $a$ , $b \in K$
$(a b) u = (a b) u$ , for any scalars $a$ , $b \in K$
$1 u = u$ , for the unit scalar $1 \in K$

Some of the most commonly used vector spaces are Polynomial Space, Matrix Space and Function Space.

Vector Subspace

Let $W$ be a subset of a vector space $V$ . Then $W$ is a subspace of $V$ if:

The zero vector $0$ belongs to $W$
For every $u$ , $v \in W$ ; $k \in K$ : a. The sum $u + v \in W$ b. The multiple $k u \in W$

Basis and Dimension

Basis

A basis of a vector space simply represents the set of mimimum independent vectors which are required to span the entire vecor space.

The number of such independent vectors is called the dimension of the vector space.

A set of vectors $S$ is called a basis of $V$ if it has the following two properties:

$S$ is linearly independent
$S$ spans $V$

Dimension

A vector space $V$ is said to be of finite dimension $n$ or $n - d i m e n s i o n a l$ , written as $d i m V = n$ , if $V$ has a basis with $n$ elements.

PYQs

Linear Dependence and Independence

1) Express basis vectors $e_{1} = (1, 0)$ and $e_{2} = (0, 1)$ as linear combination of $α_{1} = (2, - 1)$ and $α_{2} = (1, 3)$ .

[2018, 10M]

2) The vectors $V_{1} = (1, 1, 2, 4)$ , $V_{2} = (2, - 1, - 5, 2)$ , $V_{3} = (1, - 1, - 4, 0)$ and $V_{4} = (2, 1, 1, 6)$ are linearly independent. Is it true? Justify your answer.

[2015, 10M]

3) Show that the vectors $X_{1} = (1, 1 + i, i)$ , $X_{2} = (i, - i, 1 - i)$ and $X_{3} = (0, 1 - 2 i, 2 - i)$ in $C^{3}$ are linearly independent over the field of real numbers but are linearly dependent over the field of complex numbers.

[2013, 8M]

4) Show that the vectors $(1, 1, 1)$ , $(2, 1, 2)$ and $(1, 2, 3)$ are linearly independent in $R^{(3)}$ . Let $R^{(3)} \to R^{(3)}$ be a linear transformation defined by $T (x, y, z) = (x + 2 y + 3 z, x + 2 y + 5 z, 2 x + 4 y + 6 z)$ . Show that the images of above vectors under are linearly dependent. Give the reason for the same.

[2011, 10M]

5) In the space $R^{n}$ determine whether or not the set ${e_{1} - e_{2}, e_{2} - e_{3}, \dots \dots, e_{n - 1} - e_{n}, e_{n} - e_{1}}$ is linearly independent.

[2010, 10M]

6) Find the values of $k$ for which the vectors $(1, 1, 1, 1)$ , $(1, 3, - 2, k)$ , $(2, 2 k - 2, - k - 2, 3 k - 1)$ and $(3, k + 2, - 3, 2 k + 1)$ are linearly independent in $R^{4}$ .

[2005, 12M]

Vector Spaces and Subspaces

1) Find one vector in $R^{3}$ which generates the intersection of $V$ and $W$ , where $V$ is the $x y - p l a n e$ and $W$ is the space generated by the vectors $(1, 2, 3)$ and $(1, - 1, 1)$ .

[2014, 10M]

2) Let $V$ be the vector space of all $2 \times 2$ matrices over the field of real numbers. Let $W$ be the set consisting of all matrices with zero determinant. Is $W$ a subspace of $V$ ? Justify your answer?

[2012, 8M]

3) Let $S$ be a non-empty set and let $V$ denote the set of all functions from $S$ into $R$ . Show that $V$ is a vector space with respect to the vector addition $(f + g) (x) = f (x) + g (x)$ and scalar multiplication $(c \cdot f) (x) = c f (x)$ .

[2008, 12M]

4) Let $S$ be any non-empty subset of a vector space $V$ over the field $F$ . Show that the set ${a_{1} α_{1} + a_{2} α_{2} + \dots + a_{n} α_{n} : a_{1}, a_{2}, \dots, a_{n} \in F, α_{1}, α_{2}, \dots \dots, α_{n} \in S, n \in N}$ is the subspace generated by $S$ .

[2003, 12M]

Basis and Dimension

1) Suppose $U$ and $W$ are distinct four dimensional subspaces of a vector space $V$ , where $d i m V = 6$ . Find the possible dimensions of subspace $U \cap W$ .

[2017, 10M]

2) If $w_{1} = {(x, y, z) | x + y - z = 0}$ , $w_{2} = {(x, y, z) | 3 x + y - 2 z = 0}$ , $w_{3} = {(x, y, z) | x - 7 y + 3 z = 0}$ , then find $\dim (w_{1} \cap w_{2} \cap w_{3})$ and $\dim (w_{1} + w_{2})$ .

[2016, 3M]

3) Find the dimension of the subspace of $R^{4}$ , spanned by the set ${(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)}$ . Hence find its basis.

[2015, 12M]

4) Let $V$ and $W$ be the following subspaces of $R^{4} : V = {(a, b, c, d) : b - 2 c + d = 0}$ and $W = {(a, b, c, d) : a = d, b = 2 c}$ . Find a basis and the dimension of:
i. $V$
ii. $V \cap W$ .

[2014, 15M]

5) Let $V$ be an $n - d i m e n s i o n a l$ vector space and $T : V \to V$ be an invertible linear operator. If $β = {X_{1}, X_{2}, \dots X_{n}}$ is a basis of $V$ , show that $β^{'} = {T X_{1}, T X_{2}, \dots T X_{n}}$ is also a basis of $V$ .

[2013, 8M]

6) Prove or disprove the following statement: If $B = {b_{1}, b_{2}, b_{3}, b_{4}, b_{5}}$ is a basis for $R^{5}$ and $V$ is a two-dimensional subspace of $R^{5}$ , then $V$ has a basis made of two members of $B$ .

[2012, 12M]

7) Find the dimension and a basis for the space $W$ of all solutions of the following homogeneous system using matrix notation:

x_{1} + 2 x_{2} + 3 x_{3} - 2 x_{5} = 0

2 x_{1} + 4 x_{2} + 8 x_{3} + x_{4} + 9 x_{5} = 0

3 x_{1} + 6 x_{2} + 13 x_{3} + 4 x_{4} + 14 x_{5} = 0

[2012, 12M]

8) Show that the subspaces of $R^{3}$ spanned by two sets of vectors ${(1, 1, - 1), (1, 0, 1)}$ and ${(1, 2, - 3), (5, 2, 1)}$ are identical. Also find the dimension of this subspace.

[2011, 10M]

9) Prove that the set $V$ of all $3 \times 3$ real symmetric matrices forms a linear subspace of the space of all $3 \times 3$ real matrices. What is the dimension of this subspace? Find at least one of the bases for $V$ .

[2009, 20M]

10) Prove that the set $V$ of the vectors $(x_{1}, x_{2}, x_{3}, x_{4})$ in which $R^{4}$ satisfy the equation $x_{1} + x_{2} + x_{3} + x_{4} = 0$ and $2 x_{1} + 3 x_{2} - x_{3} + x_{4} = 0$ , is a subspace of $R^{4}$ . What is dimension of this subspace? Find one of its bases.

[2009, 12M]

11) Find the dimension of the subspace of $R^{4}$ spanned by the set ${(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)}$ . Hence find a basis for the subspace.

[2008, 15M]

12) Let $W$ be the set of all $3 \times 3$ symmetric matrices over $R$ . Does it from a subspace of the vector space of the $3 \times 3$ matrices over $R$ ? In case it does, construct a basis for this space and determine its dimension.

[2007, 15M]

13) Let $S$ be the vector space of all polynomials, $p (x)$ with real coefficients, of degree less than or equal to two considered over the real field $R$ such that $p (0)$ and $p (1) = 0$ . Determine a basis for $S$ and hence its dimension.

[2007, 12M]

14) Let $V$ be the vector space of all $2 \times 2$ matrices over the field $F$ . Prove that $V$ has dimension 4 by exhibiting a basis for $V$ .

[2006, 12M]

15) Let $V$ be the vector space of polynomials in $x$ of degree $\leq n$ over $R$ . Prove that the set ${1, x, x^{2}, \dots, x^{n}}$ is a basis for the set of all polynomials in $x$ .

[2005, 12M]

16) Let $S$ be space generated by the vectors ${(0, 2, 6), (3, 1, 6), (4, - 2, - 2)}$ . What is the dimension of the space $S$ ? Find a basis for $S$ .

[2004, 12M]

17) Show that the vectors $(1, 0, - 1)$ , $(0, - 3, 2)$ and $(1, 2, 1)$ form a basis for the vector space $R^{3} (R)$ .

[2001, 12M]

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