BSc CSIT (TU) Science Mathematics II (BSc CSIT, MTH163) Question Paper 2075 Nepal

Vector Space

A vector space $V$ over a field $F$ (here $F=\mathbb{R}$) is a non-empty set equipped with two operations, vector addition $(+:V\times V\to V)$ and scalar multiplication $(\cdot:F\times V\to V)$, satisfying the following axioms for all $u,v,w\in V$ and $\alpha,\beta\in F$.

Axioms

Closure

• $u+v\in V$
• $\alpha u\in V$

Additive axioms

1. Commutativity: $u+v=v+u$
2. Associativity: $(u+v)+w=u+(v+w)$
3. Additive identity: there exists $0\in V$ with $u+0=u$
4. Additive inverse: for each $u$ there exists $-u\in V$ with $u+(-u)=0$

Scalar axioms 5. Distributivity over vector addition: $\alpha(u+v)=\alpha u+\alpha v$ 6. Distributivity over scalar addition: $(\alpha+\beta)u=\alpha u+\beta u$ 7. Associativity of scalars: $\alpha(\beta u)=(\alpha\beta)u$ 8. Multiplicative identity: $1\cdot u=u$

The set of all $2\times2$ real matrices $M_{2}(\mathbb{R})$

Let $V=M_{2}(\mathbb{R})=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}: a,b,c,d\in\mathbb{R}\right\}$ with usual matrix addition and scalar multiplication.

Closure: Sum of two $2\times2$ matrices is a $2\times2$ matrix; a scalar multiple of a $2\times2$ matrix is a $2\times2$ matrix. $\checkmark$

Commutativity & Associativity: Entrywise addition of real numbers is commutative and associative, so matrix addition is too. $\checkmark$

Additive identity: The zero matrix $0=\begin{bmatrix}0&0\\0&0\end{bmatrix}$ satisfies $A+0=A$. $\checkmark$

Additive inverse: For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, $-A=\begin{bmatrix}-a&-b\\-c&-d\end{bmatrix}\in V$ and $A+(-A)=0$. $\checkmark$

Scalar axioms (5–8): Since each entry is a real number, the distributive, associative and identity laws of $\mathbb{R}$ give

All eight axioms hold, so $M_{2}(\mathbb{R})$ is a vector space over $\mathbb{R}$ (it has dimension 4).

Linear Dependence and Independence

A set of vectors $\{v_1,v_2,\dots,v_n\}$ in a vector space $V$ is linearly independent if the only scalars $c_1,\dots,c_n$ satisfying

are $c_1=c_2=\cdots=c_n=0$. If a non-trivial solution exists, the set is linearly dependent.

Worked Example

Test $v_1=(1,2,3),\ v_2=(2,1,0),\ v_3=(3,3,3)$ in $\mathbb{R}^3$.

Form the matrix with these as rows and reduce:

A zero row appears, so the vectors are linearly dependent. Indeed $v_3=v_1+v_2$.

Basis and Dimension of the spanned subspace

The non-zero rows of the echelon form are independent, so a basis of $W=\text{span}\{v_1,v_2,v_3\}$ is

Hence $\dim W = 2$ (the rank of the matrix). The subspace is a plane through the origin in $\mathbb{R}^3$.

Linear Transformation

Let $V$ and $W$ be vector spaces over a field $F$. A map $T:V\to W$ is a linear transformation if for all $u,v\in V$ and scalar $\alpha\in F$:

Equivalently, $T(\alpha u+\beta v)=\alpha T(u)+\beta T(v)$.

Example: verify linearity

Let $T:\mathbb{R}^2\to\mathbb{R}^2$ be $T(x,y)=(x+2y,\ 3x-y)$.

Take $u=(x_1,y_1),\ v=(x_2,y_2)$ and scalars $\alpha,\beta$. Then $\alpha u+\beta v=(\alpha x_1+\beta x_2,\ \alpha y_1+\beta y_2)$ and

Hence $T$ is linear.

Matrix representation (standard bases)

Apply $T$ to the standard basis $e_1=(1,0),\ e_2=(0,1)$:

These become the columns of the matrix:

Check: $[T]\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+2y\\3x-y\end{bmatrix}$, matching $T(x,y)$. $\checkmark$

A vector space over a field $F$ is a set $V$ with vector addition and scalar multiplication satisfying the eight axioms (closure under both operations, commutativity and associativity of addition, additive identity $0$ and inverses, the two distributive laws, associativity of scalars, and $1\cdot v=v$).

Examples:

1. $\mathbb{R}^n$ — the set of all real $n$-tuples with componentwise addition and scalar multiplication.
2. $P_n$ — the set of all polynomials of degree $\le n$ with real coefficients, under ordinary polynomial addition and scalar multiplication. (Also valid: $M_{m\times n}(\mathbb{R})$, the space of $m\times n$ real matrices.)

For $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$, the determinant is

so $A$ is invertible. Using $A^{-1}=\dfrac{1}{\det A}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$:

Check: $A A^{-1}=\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}-2&1\\1.5&-0.5\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.\ \checkmark$

A linear transformation is a map $T:V\to W$ between vector spaces over the same field $F$ that preserves the vector-space operations; that is, for all $u,v\in V$ and every scalar $\alpha\in F$:

Equivalently, $T(\alpha u+\beta v)=\alpha T(u)+\beta T(v)$. A consequence is $T(0_V)=0_W$. Example: $T:\mathbb{R}^2\to\mathbb{R}^2,\ T(x,y)=(x+y,\ x-y)$.

Rank–Nullity Theorem

Let $T:V\to W$ be a linear transformation where $V$ is a finite-dimensional vector space. Then

where $\operatorname{rank}(T)=\dim(\text{Im }T)$ is the dimension of the image (range) and $\operatorname{nullity}(T)=\dim(\ker T)$ is the dimension of the kernel (null space).

For an $m\times n$ matrix $A$ representing $T$, this reads $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$ (number of columns).

An orthonormal set of vectors $\{v_1,v_2,\dots,v_n\}$ in an inner-product space is a set that is both orthogonal and normalized, i.e.

This means every pair of distinct vectors is orthogonal ($\langle v_i,v_j\rangle=0$ for $i\neq j$) and each vector is a unit vector ($\|v_i\|=1$).

Example: the standard basis $\{(1,0,0),(0,1,0),(0,0,1)\}$ of $\mathbb{R}^3$ is orthonormal.

For a square matrix $A$ of order $n$, the characteristic equation is

where $\lambda$ is a scalar and $I$ is the $n\times n$ identity matrix. The left side, $p(\lambda)=\det(A-\lambda I)$, is the characteristic polynomial (a degree-$n$ polynomial in $\lambda$). Its roots are the eigenvalues of $A$.

Example: for $A=\begin{bmatrix}2&1\\1&2\end{bmatrix}$, $\det(A-\lambda I)=(2-\lambda)^2-1=\lambda^2-4\lambda+3=0$, giving eigenvalues $\lambda=1,3$.

A subspace $W$ of a vector space $V$ over a field $F$ is a non-empty subset $W\subseteq V$ that is itself a vector space under the operations of $V$. Equivalently, $W$ is a subspace iff:

1. $0\in W$ (it contains the zero vector),
2. $W$ is closed under addition: $u,v\in W\Rightarrow u+v\in W$,
3. $W$ is closed under scalar multiplication: $\alpha\in F,\ u\in W\Rightarrow \alpha u\in W$.

Example: $W=\{(x,y,0):x,y\in\mathbb{R}\}$, the $xy$-plane, is a subspace of $\mathbb{R}^3$ — it contains $(0,0,0)$ and is closed under addition and scalar multiplication.

The trace of a square matrix is the sum of its main-diagonal (top-left to bottom-right) entries. For

the diagonal entries are $1,5,9$, so

Cayley–Hamilton Theorem

Every square matrix satisfies its own characteristic equation. That is, if $A$ is an $n\times n$ matrix with characteristic polynomial

then substituting $A$ for $\lambda$ gives the zero matrix:

A useful consequence: if $A$ is invertible ($c_0\neq0$), then $A^{-1}$ can be expressed as a polynomial in $A$.