TIFR2021-Maths-A: 14

Question

Let $\mathbb{R}^{\mathbb{N}}$ denote the real vector space of sequences $(x_0,x_1,x_2,\dots)$ of real numbers. Define a linear transformation $T:\mathbb{R}^{\mathbb{N}}\rightarrow\mathbb{R}^{\mathbb{N}}$ by 

$$(x_0,x_1,\dots,)\mapsto (x_0+x_1,x_1+x_2,\dots).$$

Which one of the following statements is correct?

• A. The kernel of $T$ is infinite dimensional
• B. The image of $T$ is infinite dimensional
• C. The quotient vector space $\mathbb{R}^{\mathbb{N}}/T(\mathbb{R}^{\mathbb{N}})$ is infinite dimensional
• D. None of the other three statements is correct