Recent questions in Transformation properties

Transformation properties

Let $$\left\{v_{1},\ v_{2}, \dots,\ v_{n}\right\}$$ be a basis for a vector space V. Prove that if a linear transformation $$\displaystyle{T}\ :\ {V}\rightarrow\ {V}$$ satisfies $$\displaystyle{T}{\left({v}_{{{1}}}\right)}={0}\ \text{for}\ {i}={1},\ {2},\dot{{s}},\ {n},$$ then T is the zero transformation. Getting Started: To prove that T is the zero transformation, you need to show that $$\displaystyle{T}{\left({v}\right)}={0}$$ for every vector v in V. (i) Let v be an arbitrary vector in V such that $$\displaystyle{v}={c}_{{{1}}}\ {v}_{{{1}}}\ +\ {c}_{{{2}}}\ {v}_{{{2}}}\ +\ \dot{{s}}\ +\ {c}_{{{n}}}\ {v}_{{{n}}}.$$ (ii) Use the definition and properties of linear transformations to rewrite $$\displaystyle{T}\ {\left({v}\right)}$$ as a linear combination of $$\displaystyle{T}\ {\left({v}_{{{1}}}\right)}$$. (iii) Use the fact that $$\displaystyle{T}\ {\left({v}_{{i}}\right)}={0}$$ to conclude that $$\displaystyle{T}\ {\left({v}\right)}={0}$$, making T the zero tranformation.

Transformation properties

Transformation properties

Whether the function is a linear transformation or not. $$\displaystyle{T}\ :\ {R}^{{{2}}}\rightarrow{R}^{{{3}}},{T}{\left({x},{y}\right)}={\left(\sqrt{{{x}}},{x}{y},\sqrt{{{y}}}\right)}$$

Transformation properties

To determine. The correct graph for the function$$\displaystyle{\left[{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}\right.}$$ is B

Transformation properties

To prove: $$\displaystyle{\left[{a},\ {b}\right]}\ {>}\ {\left[{c},\ {d}\right]}\ \text{if and only if}\ {a}{b}{d}^{{2}}\ -\ {c}{d}{b}^{{{2}}}\ \in{D}^{+}$$. Given information: $$\displaystyle\text{Acoording to the definition of "greater than,"}\ {>}\ \text{is defined in Q by}\ {\left[{a},\ {b}\right]}\ {>}\ {\left[{c},\ {d}\right]}\ \text{if and only if}\ {\left[{a},\ {b}\right]}\ -\ {\left[{c},\ {d}\right]}\in{Q}^{{+}}$$

Transformation properties

Important questions of Geometry for SSC CGL Tier I In my previous session, I have discussed some concepts related to triangles. Today I will discuss some important questions of Geometry which used to appear in SSC exams. Generally, questions asked from this section are based on properties of various shapes like lines, angles, triangles, rhombus, circles etc.

Transformation properties

$$\displaystyle\text{Let A be an}\ {n}\ \times\ {n}\ \text{matrix and suppose that}\ {L}:{M}_{{\cap}}\ \rightarrow\ {M}_{{\cap}}\ \text{is defined by}{L}{\left({x}\right)}={A}{X},\text{for}\ {X}\in{M}_{{\cap}}.\text{Show that L is a linear transformation.}$$

Transformation properties

To find: The linear transformation $$\displaystyle{\left({T}_{{2}}{T}_{{1}}\right)}{\left({v}\right)}$$ for an arbitrary vector v in V. The vectors $$\displaystyle{\left\lbrace{v}_{{1}},{v}_{{2}}\right\rbrace}$$ is vasis for the vector space V. Given: The linear transformation with satisfying equations $$\displaystyle{T}_{{{1}}}{\left({v}_{{1}}\right)}={3}{v}_{{{1}}}+{v}_{{{2}}},$$ $$\displaystyle{T}_{{{1}}}{\left({v}_{{1}}\right)}=-{3}{v}_{{{1}}}+{v}_{{{2}}},\ {T}_{{{2}}}{\left({v}_{{{1}}}\right)}=-{5}{v}_{{{2}}},$$ and $$\displaystyle{T}_{{{2}}}{\left({v}_{{2}}\right)}=-{v}_{{{1}}}+{6}{v}_{{{2}}}$$ are given as $$\displaystyle{T}_{{{1}}}\ :\ {V}\ \rightarrow{V}$$ and $$\displaystyle{T}_{{{2}}}\ :\ {V}\rightarrow{V}.$$

Transformation properties

Prove that the metric area is defined as $$\displaystyle{P}\ {<}\ {x}_{{{1}}},\ {y}_{{{1}}}\ {>}\ {\quad\text{and}\quad}\ {Q}\ {<}\ {x}_{{{2}}},\ {y}_{{{2}}}\ {>}$$. If the proof of examples says that the first properties (positive definiteness and symmetry) are trivial. Prove the versatility of properties for a given space.

Transformation properties

To show: $$\displaystyle{a}\ {<}\ {\frac{{{a}+{b}}}{{{2}}}}\ {<}\ {b}$$ Given information: a and b are real numbers.

Transformation properties

Angles A and B are vertical angles. If $$\displaystyle{m}\angle{A}={104}∘$$ what is the $$m\angle B?$$ $$\displaystyle{m}\angle{B}=∘{d}{e}{g}{r}{e}{e}{s}$$

Transformation properties

Whether T is a linear transformation, that is $$\displaystyle{T}:{C}^{{1}}{\left[-{1},{1}\right]}\rightarrow{R}^{{1}}$$ defined by $$\displaystyle{T}{\left({f}\right)}={f}'{\left({0}\right)}$$

Transformation properties

Angles A and B are complementary. If $$\displaystyle{m}\angle{A}={27}∘$$, what is the $$m\angle B?$$ $$\displaystyle{m}\angle{B}= \circ {d}{e}{g}{r}{e}{e}{s}$$

Transformation properties

Whether T is a linear transformation, that is $$\displaystyle{T}:{C}{\left[{0},{1}\right]}\rightarrow{C}{\left[{0},{1}\right]}$$ defined by $$\displaystyle{T}{\left({f}\right)}={g}$$, where $$\displaystyle{g{{\left({x}\right)}}}={e}^{{{x}}}{f{{\left({x}\right)}}}$$

Transformation properties

Whether the function is a linear transformation or not. $$\displaystyle{T}\ :\ {R}^{{{2}}}\rightarrow{R}^{{{2}}},{T}{\left({x},{y}\right)}={\left({x},{y}^{{{2}}}\right)}$$

Transformation properties

For the V vector space contains all $$\displaystyle{2}\times{2}$$ matrices. Determine whether the $$\displaystyle{T}:{V}\rightarrow{R}^{{1}}$$ is the linear transformation over the $$\displaystyle{T}{\left({A}\right)}={a}\ +\ {2}{b}\ -\ {c}\ +\ {d},$$ where $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$

Transformation properties

Give an example of an undefined term and a defined term in geometry. Explain the difference between an undefined term and a defined term.

Transformation properties

Angles A and B are supplementary. If $$\displaystyle{m}\angle{A}={78}°$$ what is $${m}\angle{B}$$

Transformation properties