Let $T: R^2 \to R^3$ be defined as $T(x, y) = (x + y, x - y, y)$ for…

Question

Let $T: R^2 \to R^3$ be defined as

$T(x, y) = (x + y, x - y, y)$ for any $x, y \in R$.

Then show that $T$ is linear. Also find a basis and dimension of its range space and null space.

Related pages

• Leibnitz Test for Alternating Series — topic
• Engineering Mathematics-II syllabus

More PYQs on Leibnitz Test for Alternating Series

• Define the terms: Absolute convergence and conditional convergence. Also verify whether the following series is conditional convergent or not? $1 - \frac{1}{\s…
• Solve the following ordinary differential equations: (a) $xydy - y^2dx = (x^2 + y^2)dy$ (b) $(D^2 - 2D + 1)y = x\sin x$
• Find the rank of the following matrix for various values of k: $\begin{bmatrix} 1 & k & -1 & 2 \\ 2 & -1 & k & 5 \\ 1 & 10 & -6 & 1 \end{bmatrix}$
• a) Test the following system of equations for consistency: $$2x - 3y + 7z = 5$$ $$3x + y - 3z = 13$$ $$2x + 19y - 47z = 32$$ b) Reduce the matrix A to normal f…
• Define $T$ on $R^3$ as $T(x, y, z) = (2y + z, x - 4y, 3x)$ for any $x, y, z \in R$ where $R$ is the set of all reals. Then show that $T$ is linear and also fin…
• a) Derive the solution for wave equation of a stretched flexible string of length L. b) Solve: $(D^2 + 2D'^2 + 3DD' - 4D - 5D' + 3)z = 0$. c) Find P.I. of $(D^…
• a) Prove that a super set of a linearly dependent set in a vector space in linearly dependent. b) Verify Rank-Nullity theorem for the linear transformation $T…
• (a) Show that the series $\frac{1}{\sqrt{3}\cdot 4} + \frac{1}{\sqrt{4}\cdot 5} + \frac{1}{\sqrt{5}\cdot 6} + \dots$ is divergent. (b) Test whether the given s…
• a) Solve: $r - 2s + t = \sin(x-y)$. b) Find P.I. of $(D^2 - D'^2 - 2D + 2D')z = xy$. c) Using the method of separation of variables derive the solution for one…
• a) Define $T : R^2 \to R^2$ as $T(x,y) = (2x - 3y, x+y)$ for any $x,y \in R$. Then show that T is linear and also find the matrix associated with T relative to…