  How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Consider the vector space V over $\mathbb{R}$ of the polynomial functions of degree less than or equal to 3 defined on $\mathbb{R}$. Let $T : V \longrightarrow V$ defined by \$latex (Tf)(x) = f(x)-xf'(x). Then the rank of T is  (a) 1  (b) 2 (c) 3 (d) 4 [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]IIT JAM 2018 Problem 9[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.1" open="off"]Vector Space [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.1" open="off"]Abstract Algebra By S.K Mapa[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="4.1" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0"][et_pb_tab title="Hint 0" _builder_version="4.1" hover_enabled="0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.1" hover_enabled="0"]Rank(T) = dim(Range(T)) There is one easy way to calculate rank of every linear transformation. Step 1:  Take by basis $\beta= \{e_1,....,e_n\}$ of the vector space $V$. Step 2: Write down the matrix $[T]_{\beta}^{\beta}$ Step 3: Calculate the rank of the matrix $[T]_{\beta}^{\beta}$ Now can you follow these steps to get the answer?  [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.1" hover_enabled="0"]Standard Basis of $V$ is $\{1,x,x^{2},x^{3}\} = \beta$ $(Tf) (x) =f(x) - xf^{'}(x)$ $(T1) (x) = 1 - 0 = 1$; $(Tx) (x) = x - x = 0$; $(T x^{2}) (x)= x^{2} - 2x^{2} = -x^{2}$ ; $(T x^{3}) (x) = -2x^{3}$ So, $[T]_{\beta}^{\beta} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -2 \\ \end{pmatrix}$ Hence the rank is $3$[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

# Watch the video

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# Similar Problems

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