We will cover following topics

Kernel and Image
- Kernel
- Image
Rank and Nullity
Matrix of a Linear Transformation
- Change of Basis
PYQs

Kernel and Image

Kernel

Let $A$ be any $m \times n$ matrix over a field $K$ viewed as a linear map $A : K^{n} \to K^{m}$ . Then, kernel of $A$ , denoted by $K e r (A)$ , is the null space of $A$ , that is,

$K e r (A) = n u l l s p (A)$ , and

Here, $n u l l s p (A)$ is the solution space of homogeneous system $A X = 0$ .

Image

The image of a linear transformation or matrix is the span of the vectors of the linear transformation, that is,

$I m A = c o l s p (A)$

Rank and Nullity

Rank

The rank of a matrix is defined as:

The maximum number of linearly independent column vectors in the matrix or
The maximum number of linearly independent row vectors in the matrix.

Both of the above definitions are equivalent and give the same answer.

For a $m \times n$ matrix, rank of the matrix is less than or equal to the minumum of $m$ and $n$ .

Nullity

The null space of a linear mapping is the set of vectors in the domain of the mapping which are mapped to the zero vector. It is also called as kernel of the the mapping. Nullity represnts the number of vectors in the null space.

For a linear mapping $A$ :

$R a n k (A) = d i m (I m A)$ ,
$n u l l i t y (A) = d i m (K e r F)$ .

Rank-Nullity Theorem

Let $V$ be finite dimensional and let $A : V \to U$ be linear. Then,

$d i m V = d i m (K e r A) + d i m (I m A) = n u l l i t y (A) + r a n k (A)$

Matrix of a Linear Transformation

Let $T$ be a linear operator (transformation) from a vector space $V$ into itself, and suppose $S = {u_{1}, u_{1}, \dots, u_{n}}$ is a basis of $V$ .

Now, all the vectors $T (u_{1})$ , $T (u_{2})$ , $\dots$ , $T (u_{n})$ are vectors in $V$ , and so each is a linear combination of the vectors in the basis $S$ ; say,

$T (u_{1}) = a_{11} u_{1} + a_{12} u_{2} + \dots + a_{1 n} u_{n}$ ,

$T (u_{2}) = a_{21} u_{1} + a_{22} u_{2} + \dots + a_{2 n} u_{n}$ , $\dots \dots$

$T (u_{n}) = a_{n 1} u_{1} + a_{n 2} u_{2} + \dots + a_{n n} u_{n}$ .

The transpose of the above matrix of coefficients, denoted by $m_{S} (T)$ or $[T]_{s}$ , is called the matrix representation of $T$ relative to the basis $S$ , or simply the matrix of $T$ in the basis $S$ .

The subscript $S$ may be omitted if the basis $S$ is understood.

Change of Basis

Let $S = {u_{1}, u_{2}, \dots, u_{n}}$ be a basis of a vector space $V$ , and let $S^{'} = {v_{1}, v_{2}, \dots, v_{n}}$ be another basis. Because $S$ is a basis, each vector in the “new” basis $S^{'}$ can be written uniquely as a linear combination of the vectors in $S$ ; say,
$v_{1} = a_{11} u_{1} + a_{12} u_{2} + \dots + a_{1 n} u_{n}$ ,

$v_{2} = a_{21} u_{1} + a_{22} u_{2} + \dots + a_{2 n} u_{n}$ , $\dots \dots$ ,

$v_{n} = a_{n 1} u_{1} + a_{n 2} u_{2} + \dots + a_{m n} u_{n}$ .

Let $P$ be the transpose of the above matrix of coefficients; that is, let $P = [p_{i j}]$ , where $p_{i j} = a_{j i}$ .

Then $P$ , is called the change-of-basis matrix (or transition matrix) from the “old” basis $S$ to the “new” basis $S^{'}$ .

PYQs

Kernel and Image

1) Consider the matrix mapping $A : R^{4} \to R^{3}$ , where $A = [\begin{matrix} 1 & 2 & 3 & 1 \\ 1 & 3 & 5 & - 2 \\ 3 & 8 & 13 & - 3 \end{matrix}]$ . Find a basis and dimension of the image of $A$ and those of the kernel $A$ .

[2017, 15M]

2) Let $T : R^{3} \to R^{3}$ be the linear transformation defined by $T (α, β, γ) = (α + 2 β - 3 γ, 2 α + 5 β - 4 γ, α + 4 β + γ$ . Find a basis and the dimension of the image of $T$ and the kernel of $T$ .

[2012, 12M]

3) Let $T$ be the linear transformation from $R^{3}$ to $R^{4}$ defined by $T (x_{1}, x_{2}, x_{3})$ = $(2 x_{1} + x_{2} + x_{3}, x_{1} + x_{2}, x_{1} - x_{3}, 3 x_{1} + x_{2} - 2 x_{3})$ for each $(x_{1}, x_{2}, x_{3}) \in R^{3}$ . Determine a basis for the Null space of $T$ . What is the dimension of the Range space of $T$ ?

[2007, 12M]

4) Show that $f : R^{3} \to R$ is a linear transformation, where $f (x, y, z) = 3 x + y - z$ . What is the dimension of the Kernel? Find a basis for the Kernel.

[2004, 12M]

5) Let $T$ : $R^{5} \to R^{5}$ be a linear mapping given by $T (a, b, c, d, e) = (b - d, d + e, b, 2 d + e, b + e)$ . Obtain bases for its null space and range space.

[2002, 15M]

Rank and Nullity

1) Find the nullity and a basis of the null space of the linear transformation $A : R^{(4)} \to R^{(4)}$ given by the matrix $A = [\begin{matrix} 0 & 1 & - 3 & - 1 \\ 1 & 0 & 1 & 1 \\ 3 & 1 & 0 & 2 \\ 1 & 1 & - 2 & 0 \end{matrix}]$ .

[2011, 10M]

2) What is the null space of the differentiation transformation $\frac{d}{d x} : p_{n} \to p_{n}$ where $p_{n}$ is the space of all polynomials of degree $\leq n$ over the real numbers? What is the null space of the second derivative as a transformation of? What is the null space of the $k^{t h}$ derivative $p_{n}$ ?

[2010, 12M]

3) Let $L : R^{4} \to R^{3}$ be a linear transformation defined by $L (x_{1}, x_{2}, x_{3}, x_{4}) = (x_{3} + x_{4} - x_{1} - x_{2}, x_{3} - x_{2}, x_{4} - x_{1})$ . Then find the rank and nullity of $L$ . Also, determine null space and range space of $L$ .

[2009, 20M]

Matrix of a Linear Transformation

1) If $M_{2} (R)$ is space of real matrices of order $2 \times 2$ and $P_{2} (x)$ is the space of real polynomials of degree at most 2, then find the matrix representation of $T : M_{2} (R) \to P_{2} (x)$ such that $T ([\begin{matrix} a & b \\ c & d \end{matrix}]) = a + c + (a - d) x + (b + c) x^{2}$ , with respect to the standard bases of $M_{2} (R)$ and $P_{2} (x)$ , further find null space of $T$ .

[2016, 10M]

2) If $T : P_{2} (x) \to P_{3} (x)$ is such that $T (f (x)) = f (x) + 5 \int_{0}^{x} f (t) d t$ , then choosing ${1, 1 + x, 1 - x^{2}}$ and ${1, x, x^{2}, x^{3}}$ as bases of $P_{2} (x) $ a n d $ P_{3} (x)$ respectively, find the matrix of $T$ .

[2016, 6M]

3) If $A = [\begin{matrix} 1 & - 1 & 2 \\ - 2 & 1 & - 1 \\ 1 & 2 & 3 \end{matrix}]$ is the matrix representation of a linear transformation $T : P_{2} (x) \to P_{2} (x)$ with respect to the bases ${1 - x, x (1 - x), x (1 + x)}$ and ${1, 1 + x, 1 + x^{2}}$ then find $T$ .

[2016, 6M]

4) Let $V = R^{3}$ and $T \in A (V)$ , for all $a_{i} \in A (V)$ , be defined by $T (a_{1}, a_{2}, a_{3}) = (2 a_{1} + 5 a_{2} + a_{3}, - 3 a_{1} + a_{2} - a_{3}, a_{1} + 2 a_{2} + 3 a_{3})$ . What is the matrix $T$ relative to the basis $V_{1} = (1, 0, 1)$ , $V_{2} = (- 1, 2, 1)$ , $V_{3} = (3, - 1, 1)$ ?

[2015, 12M]

5) Let $P_{n}$ denote the vector space of all real polynomials of degree at most $n$ : $P_{2} \to P_{3}$ be linear transformation given by $T (f (x)) = \int_{0}^{x} p (t) d t$ , $p (x) \in P_{2}$ . Find the matrix of $T$ with respect to the bases ${1, x, x^{2}}$ and ${1, x, 1 + x^{2}, 1 + x^{3}}$ of $P_{2}$ and $P_{3}$ respectively. Also find the null space of $T$ .

[2013, 10M]

6) Consider the linear mapping $f : R^{2} \to R^{2}$ by $f (x, y) = (3 x + 4 y, 2 x - 5 y)$ . Find the matrix $A$ relative to the basis ${(1, 0), (0, 1)}$ and the matrix $B$ relative to the basis ${(1, 2), (2, 3)}$ .

[2012, 12M]

7) Let $M = [\begin{matrix} 4 & 2 & 1 \\ 0 & 1 & 3 \end{matrix}]$ . Find the unique linear transformation $T : R^{3} \to R^{2}$ so that $M$ is the matrix of $T$ with respect to the basis $β = {v_{1} = (1, 0, 0) v_{2} = (1, 1, 0) v_{3} = (1, 1, 1)}$ of $R^{3}$ and $β^{'} = {w_{1} = (1, 0), w_{2} = (1, 1)}$ of $R^{2}$ . Also find $T (x, y, z)$ .

[2010, 20M]

8) Let $T$ be a linear transformation from a vector space $V$ over reals into $V$ such that $T - T^{2} = I$ . Show that $T$ is invertible.

[2010, 10M]

9) Let $β = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}$ and $β^{'} = {(2, 1, 1), (1, 2, 1), (- 1, 1, 1)}$ be the two ordered bases of $R^{3}$ . Then, find a matrix representing the linear transformation $T : R^{3} \to R^{3}$ which transforms $β$ into $β^{'}$ . Use this matrix representation to find $T (x)$ , where $x = (2, 3, 1)$ .

[2009, 20M]

10) Show that $B = {(1, 0, 0), (1, 1, 0), (1, 1, 1)}$ is a basis of $R^{3}$ . Let $T : R^{3} \to R^{3}$ be a linear transformation such that $T (1, 0, 0) = (1, 0, 0)$ , $T (1, 1, 0) = (1, 1, 1)$ and $T (1, 1, 1) = (1, 1, 0)$ . Find $T (x, y, z)$ .

[2008, 15M]

11) Consider the vector space $X := {p (x), {p (x)$ is a polynomial of degree less than or equal to 3 with real coefficients $}$ over the real field $R$ . Define the map $D : X \to X$ by
$(D p) (x) := p_{1} + 2 p_{2} x + 3 p_{3} x^{2}$
where $p (x) := p_{0} + p_{1} x + p_{2} x^{2} - p_{3} x^{3}$ .
Is $D$ a linear transformation on $x$ ? If it is, then construct the matrix representation for $D$ with respect to the ordered basis ${1, x, x^{2}, x^{3}}$ for $X$ .

[2007, 15M]

12) If $T : R^{2} \to R^{2}$ is defined by $T (x, y) = (2 x - 3 y, x + y)$ , compute the matrix of $T$ relative to the basis $β {(1, 2), (2, 3)}$ .

[2006, 15M]

13) Let $T$ be a linear transformation on $R^{3}$ whose matrix relative to the standard basis of $R^{3}$ is $[\begin{matrix} 2 & 1 & - 1 \\ 1 & 2 & 2 \\ 3 & 3 & 4 \end{matrix}]$ . Find the matrix of $$ T$ relative to the basis $β = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}$

[2005, 15M]

14) Show that the linear transformation form $R^{3}$ to $R^{4}$ which is represented by the matrix $[\begin{matrix} 1 & 3 & 0 \\ 0 & 1 & - 2 \\ 2 & 1 & 1 \\ - 1 & 1 & 2 \end{matrix}]$ is one-to-one. Find a basis for its image.

[2004, 12M]

15) Show that the mapping $T : R^{3} \to R^{3}$ , where $T (a, b, c) = (a - b, b - c, a + c)$ , is linear and non singular.

[2002, 12M]

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