Previous Year Questions on Linear Transformations

Linear Transformations - Previous Year Questions

Previous Year Questions on Linear Transformations

Previous year questions on Linear Transformations highlight their fundamental role in mathematics and its applications. Linear transformations, introduced by 19th-century mathematicians, form the basis of Linear Algebra Notes. These concepts are widely used in computer graphics, machine learning, and quantum mechanics.

Vidyasagar University

2023-24 (CBCS)

Show that the linear mapping \(T:\mathbb{R}^{3}\to \mathbb{R}^{3}\), defined by \(T(x,y,z)=(x-y,x+2y,y+3z) \), \((x,y,z)\in \mathbb{R}^{3}\) is non-singular. [2]
Show that the mapping \(T:\mathbb{R}^{3}\to \mathbb{R}^{2}\), defined by \(T(x,y,z)=(3x-2y+z,x-3y-2z) \) is a linear mapping. [2]

2021-22 (CBCS)

Let \(C[0,1] \) be the set of all real continuous functionsn on the closed interval \(C[0,1] \) and \(T\) be a mapping from \(C[0,1] \) to \(\mathbb{R} \) defined by \(T(f)=\int_{0}^{1} f(x)~dx, ~f\in C[0,1] \). Show that \(T\) is a linear transformation. [4]

2020-21 (CBCS)

Find eigen values and basis of each eigen space for the operator \(T:\mathbb{R}^{3} \to \mathbb{R}^{3} \) defined by \(T(x,y,z)=(2x+y,y-z, 2y+4z) \). [6]

2019-20 (CBCS)

A linear mappings \(T:\mathbb{R}^{3}\to \mathbb{R}^{2} \) is defined by \(T=(x_{1},x_{2}),x_{3})=(3x_{1}-2x_{2}+x_{3},x_{1}-3x_{2})-2x_{3}),~(x_{1},x_{2}),x_{3})\in \mathbb{R}^{3} \). Find the matrix \(T\) relative to the ordered bases \(\set{(0,1,1),(1,0,1),(1,1,0) } \) of \( \mathbb{R}^{3}\) and \(\set{(1,0),(0,1)} \) of \( \mathbb{R}^{2}\). [2]

2018-19 (CBCS)

No Questions

2017-18 (CBCS)

If \( A=\begin{pmatrix} \frac{1}{v_{2}} & -\frac{1}{v_{2}} \\ \frac{1}{v_{2}} & \frac{1}{v_{2}} \end{pmatrix}\), \(X=\left(x_{1},x_{2} \right)^{T} \) and \(Y=\left(y_{1},y_{2} \right)^{T} \). Verify by means of the transformation \(X=AY \) that \(x_{1}^{2}+x_{2}^{2} \) is transformed to \(y_{1}^{2}+y_{2}^{2} \). [3]

FAQs

What is a linear transformation?
A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication.
How are linear transformations represented?
They are typically represented using matrices in standard or other bases.
What is the importance of linear transformations?
They simplify complex mathematical problems and have applications in fields like computer science and physics.
What are null space and range?
Null space is the set of vectors mapped to zero, while the range is the set of all possible outputs.
What are examples of linear transformations?
Rotations, translations, scalings, and projections are examples of linear transformations.
How are linear transformations used in computer graphics?
They are used to perform transformations like scaling and rotating images.
What is the difference between linear transformations and general functions?
Linear transformations strictly preserve vector addition and scalar multiplication, unlike general functions.
What is the determinant’s role in linear transformations?
The determinant indicates whether a transformation preserves orientation or volume in a vector space.
What is the relationship between eigenvalues and linear transformations?
Eigenvalues and eigenvectors describe how a linear transformation acts on specific vectors.
Can a linear transformation be invertible?
Yes, a linear transformation is invertible if its determinant is non-zero.