Previous Year Questions on Vector Space

Vector Space - Previous Year Questions

Previous Year Questions on Vector Space

Previous year questions on Vector Space help students understand the fundamental concept of linear algebra, introduced in the 19th century. Vector spaces form the basis for various applications in mathematics, physics, and computer science, including machine learning and quantum mechanics. Explore the foundational concepts with Linear Algebra Notes.

Vidyasagar University

2023-24 (CBCS)

Let \(S\) be the subset of defined by \(S=\set{ (x,y,z)\in \mathbb{R}^{3}: x+y+z=5} \). Examine whether \(s\) is a subspace of \(\mathbb{R}^{3}\). [2]
Find the co-ordinate vector of \(\alpha\) in \(\mathbb{R}^{3} \) relative to the basis \( (\alpha_{1},\alpha_{2},\alpha_{3}) \) where \(\alpha=(2,3,3) \), \(\alpha_{1}=(2,1,1) \), \(\alpha_{2}=(1,2,1) \), \(\alpha_{3}=(1,1,2) \). [2]
If \(\alpha, \beta, \gamma \) be a basis of a vector space \(\mathbb{R}^{3} \), then show that \(\set{\alpha+\beta, \beta+ \gamma, \gamma+\alpha } \) is also a basis set for \(\mathbb{R}^{3} \). [2]
Show that \(S=\set{ (x,y,z)\in \mathbb{R}^{3}: x+y-z=0} \) is a subspace of \(\mathbb{R}^{3} \). Find the basis and dimension of this subspace. [5]
Prove that the solutions of a homogeneous system \( AX=O\) in \(n\) unknowns where \(A\) is an \(m\times n \) matrix over a field \(F\), form a subspace of \(V_{n}(F) \). [5]

2021-22 (CBCS)

Let \(V\) be a real vector space with a basis \(\set{\alpha_{1},\alpha_{2},…,\alpha_{n} } \). Examine if \(\set{\alpha_{1}+\alpha_{2},\alpha_{2}+\alpha_{3},…,\alpha_{n}+\alpha_{1} } \) is also a basis of \(V\). [5]
Find \(k\in \mathbb{R} \) so that the set \(S=\set{(1,2,1),(k,3,1),(2,k,0) } \) is linearly dependent in \(\mathbb{R}^{3} \). [3]
Find the dimension of the subspace \(S\cap T \) of \(\mathbb{R}^{4}\) where \(S=\set{(x,y,z,w)\in \mathbb{R}: x+y+z+w=0 } \) and \(T=\set{(x,y,z,w)\in \mathbb{R}: 2x+y-z+w=0 } \). [5]
Find the basis for the the column space of the matrix \( \begin{pmatrix} 1 & 2 & -1 \\ 2 & 3 & 0 \\ 1 & 1 & 1 \end{pmatrix} \). [5]
Show that the set of all points on the line \(y=mx \) forms a subspace of the vector space \(\mathbb{R}^{2} \). [5]

2020-21 (CBCS)

Examine if the set \(S=\set{ (x,y,z)\in \mathbb{R}^{3}: x^{2}+y^{2}=z^{2}} \) is a subspace of \(\mathbb{R}^{3} \). [2]
Prove that the vector space \(P\) of all real polynomials is infinite dimensional. [2]
Define a basis of a vector space. Prove that the dimension of vector space is unique. [2]
Prove that \(V\) is the vector space of polynomials in \(x\) of degree \(\leq n \) over \(\mathbb{R}\). Show that the set \(S=\set{1,x,x^{2},…,x^{n} } \) is a basis of \(V\). [6]

2019-20 (CBCS)

Let \(V\) be a vector space with \(\set{\alpha, \beta, \gamma } \) as a basis. Prove that the set \(\set{\alpha+\beta+\gamma, \beta+ \gamma, \gamma } \) is also a basis. [2]
Show that the intersection of two subspaces of a vector space over a field \(F\) is a subspace of \(V\). [3]
If \(\alpha=(1,2,2), \beta=(0,2,1), \gamma=(2,2,4) \) determine whether \( \alpha\) is a linear combination of \( \beta\) and \( \gamma\). [2]
Let \(S=\set{ (x,y,z)\in \mathbb{R}^{3}: x+y+z=0} \). Prove that \(S\) is a subspace of \(\mathbb{R}^{3} \). Find a basis of \(S\). Determine the dimension of \(S\). [4]

2018-19 (CBCS)

Let \(x,y,z\) be elements of a vector space \(V\) over a field \(F\) and let \(a,b\in F\). Show that \(\set{x,y,z}\) are linearly dependent, if \(\set{x+ay+bz,y,z}\) are linearly dependent. [2]
Let \(S=\set{ (x,y,z,w)\in \mathbb{R}^{4}: x+2y-z=0, 2x+y+w=0, } \). Prove that \(S\) is a subspace of \(\mathbb{R}^{4} \). Find a basis of \(S\). Determine the dimension of \(S\). [2+2+1]
Let \(V\) be a vector space over a field \(F\) and let \( \alpha, \beta \in V\). Then prove that the set \(W=\set{c\alpha+d\beta: c,d\in F } \) form a subspace of \(V\). If \(\alpha=(1,2,3), \beta=(3,1,0), \gamma=(2,1,3) \) then examine for \(\gamma\in W \) or not. [5]

2017-18 (CBCS)

Determine \(k\) so that the set \(\set{(1,2,1), (k,3,1), (2,k,0) } \) is lnearly independent. [2]
Find the dimension of the subspace \(\mathbb{R}^{3} \) defined by \(S=\set{ (x,y,z)\in \mathbb{R}^{3}: x+2y=z, 2x+3z=y} \). [2]

FAQs

What is a Vector Space?
A vector space is a set of vectors that can be added together and multiplied by scalars, following specific axioms.
What are subspaces in Vector Space?
Subspaces are subsets of a vector space that themselves form a vector space under the same operations.
What is the basis of a Vector Space?
The basis is a set of linearly independent vectors that span the entire vector space.
What is the dimension of a Vector Space?
The dimension is the number of vectors in the basis of the vector space.
How are Vector Spaces used in Linear Transformations?
Linear transformations map one vector space to another while preserving vector addition and scalar multiplication.
What are real-life applications of Vector Spaces?
Vector spaces are used in fields like computer graphics, machine learning, and quantum mechanics.
What is the significance of linear independence in Vector Spaces?
Linear independence ensures that no vector in a set can be represented as a combination of others.
What are orthogonal vectors in a Vector Space?
Orthogonal vectors have a dot product of zero, indicating they are perpendicular.
How are basis and dimension related in Vector Spaces?
The dimension of a vector space is defined by the number of vectors in its basis.
Why are Vector Spaces important in Linear Algebra?
Vector spaces provide the framework for understanding linear equations, transformations, and advanced mathematical concepts.