HPAS Maths Optional PYQs: Linear Algebra

On that page, you will find year-wise and question-wise solutions.

Based on the sources provided, here are the year-wise questions pertaining to Linear Algebra from the HPAS Maths Optional Question Paper-1 examinations:

HPAS 2024 Linear Algebra Questions

Question 1(a): Show that similar matrices have the same minimal polynomial.

Question 2(a): Show that the inner product space is a normed vector space but the converse is not true.

Question 4(a): Show that every non-zero finite-dimensional inner product space has an orthonormal basis.

Question 4(b): Using the concept of diagonalizability, determine $A^5$ where $$A = \begin{pmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{pmatrix}$$

Question 8(a): Show that every real n-dimensional vector space is isomorphic to $\mathbb{R}^n$.

HPAS 2023 Linear Algebra Questions

Question 1(d): Give an example of a diagonalizable matrix that does not have distinct eigen values.

Question 2(a): Using the Cauchy-Schwarz inequality, show that the cosine of an angle has an absolute value of at most 1.

Question 2(b): Let V be a finite-dimensional vector space and W be a subspace of V. Show that $\dim A(W) = \dim V – \dim W$, where $A(W)$ is the annihilator of W.

Question 3(c): Let $T: V \to W$ be a linear transformation. Then show that $\text{Rank}(T) + \text{Nullity}(T) = \dim(V)$.

Question 7(a): Let V be the vector space of real-valued functions $y=f(x)$ satisfying $\frac{d^3y}{dx^3} – 6\frac{d^2y}{dx^2} + 11\frac{dy}{dx} – 6y = 0$. Then show that V is a 3-dimensional vector space over $\mathbb{R}$.

HPAS 2021 Linear Algebra Questions

Question 1(a): Show that two finite-dimensional vector spaces over a field $F$ are isomorphic if and only if they have the same dimension.

Question 2(a): Let V be a non-zero inner product space of dimension $n$. Then show that V has an orthonormal basis.

Question 2(b): Let V be the space of all real-valued continuous functions. Define a linear operator $T: V \to V$ by $(Tf)(x) = \int_{0}^{x} f(t) dt$. Show that T has no eigenvalues.

HPAS 2020 Linear Algebra Questions

Question 1(b): Find a linear transformation $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that its image space is the plane $x+y+z=0$.

Question 2(a): Define a basis of a vector space. Find a basis of the subspace of the vector space $\mathbb{R}^4(\mathbb{R})$ generated by the subset: $\{(1,1,0,-1), (2,4,6,0), (-2,-3,-3,1), (-1,-2,-2,2), (4,6,4,-6)\}$.

Question 2(b): Let $$A = \begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}$$ Find an invertible $2 \times 2$ matrix P such that $PAP^{-1}$ is a diagonal matrix.

HPAS 2018 Linear Algebra Questions

Question 1(b): How many solutions does the following system of linear equations have? $$\begin{cases} -x+5y = -1 \\ x-y = 2 \\ x+3y = 3 \end{cases}$$

Question 5(a): Let U and V be vector spaces and $T:U \to V$ be a surjective linear mapping. If $\dim U=6$ and $\dim V=3$, find $\dim(\text{Ker } T)$.

Question 5(b): Find the rank of the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x,y,z)=(y,0,z)$.

Question 6(a): Let A be a $3\times3$ square matrix with eigenvalues 1, -1, and 0. Find the value of $\det(I+A^{100})$.

Question 6(b): Let $\mathbb{R}^4(\mathbb{R})$ be a vector space and let S be its subspace spanned by the vectors (1,2,3,0), (2,3,0,1), and (3,0,1,2). Find the dimension of the quotient space $\mathbb{R}^4/S$.

Question 7(a): In a finite-dimensional inner product space V, let $\{w_1, w_2, \dots, w_n\}$ be an orthonormal subset of V such that $\sum_{i=1}^{n} |\langle w_i, v \rangle|^2 = ||v||^2$ for all $v \in V$. Find a basis of V.

Question 7(c): Let $V=\mathbb{R}^2$ be a finite-dimensional standard inner product space. Prove that ${(-1, 0), (0, -1)}$ forms an orthonormal basis of V.

HPAS 2017 Linear Algebra Questions

Question 1(a): If A is a nilpotent matrix of index 2, show that $A(I \pm nA) = A$ for any positive integer n.

Question 2(a): Find the matrix representation of a linear transformation t on $V_3(\mathbb{R})$ defined as $t(x,y,z)=(2y+z, x-4y, 3x)$ corresponding to the basis $B = \{(1,0,0), (0,1,0), (0,0,1)\}$.

Question 2(b): State and prove the Cauchy-Schwarz inequality.

HPAS 2016 Linear Algebra Questions

Question 1(a): Determine all values of d for which rank of the matrix $$\begin{pmatrix} d & -1 & 0 & 0 \\ 0 & d & -1 & 0 \\ 0 & 0 & d & -1 \\ -6 & 11 & -6 & 1 \end{pmatrix}$$ is equal to 3.

Question 1(b): Show that 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 dependent in $\mathbb{R}^4$.

Question 2(a): Let T be a linear operator on $\mathbb{R}^3$ defined by $T(x,y,z)=(3x, x-y, 2x+y+z)$. Show that T is invertible and determine $T^{-1}$.

Question 2(b): Determine the value of a and b so that the system of equations: $$\begin{cases} 2x+3y+5z=9 \\ 7x+3y-2z=8 \\ 2x+3y+az=b \end{cases}$$ has: (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

Question 8(a): Let V be a finite dimensional inner product space over field F, and let $g: V \to F$ be a linear transformation. Then show that there exists a unique vector $y \in V$ such that $g(x) = \langle x,y \rangle$ for all $x \in V$.

HPAS 2015 Linear Algebra Questions

Question 1(a): Find characteristic equation and roots of the matrix: $$A=\begin{bmatrix}8&-6&2\\ -6&7&-4\\ 2&-4&3\end{bmatrix}$$

Question 1(b): Prove that the set S = \{(1, 2, 1), (2, 1, 0), (1, -1, 2)\} forms a basis of vector space $V_{3}(R)$.

Question 2(a): Apply matrix theory to solve the following system of equations: $$\begin{cases} x+y+z=6 \\ x-y+z=2 \\ 2x+2y-z=1 \end{cases}$$

Question 2(b): Prove that the mapping $f:V_{3}(R)\rightarrow V_{2}(R)$ defined by $f(u_{1},u_{2},u_{3})=(u_{1}-u_{2},u_{1}-u_{3})$ is a linear transformation.

Question 2(c): State and prove Cauchy-Schwarz inequality.

HPAS 2014 Linear Algebra Questions

Question 1(a): Find rank of the matrix: $$A=\begin{bmatrix}1&1&1&-1\\ 1&2&3&4\\ 3&4&5&2\end{bmatrix}$$

Question 1(b): Show that the set: $W={(a,b,c):a-3b+4c=0;a,b,c\in R}$, is a subspace of the vector space $V_{3}(R)$.

Question 2(a): Find the characteristic roots and characteristic vectors of the following matrix: $$\begin{bmatrix}1&2&3\\ 0&-4&2\\ 0&0&7\end{bmatrix}$$

Question 2(b): Solve the following equations using matrix method: $$\begin{cases} 2x_{1}+3x_{2}+x_{3}=9 \\ x_{1}+2x_{2}+3x_{3}=6 \\ 3x_{1}+x_{2}+2x_{3}=8. \end{cases}$$

Question 2(c): Prove that the mapping: $t:V_{2}(R)\rightarrow V_{3}(R)$, which is defined by $t(x,y)=(x,y,0)$ is a linear transformation from vector space $V_{2}(R)$ to the vector space $V_{3}(R)$.