Linear Transformations

A linear transformation from a vector space V to another (or the same) vector space W is a function F from V into W such that (1.1)<math display='block'> <mrow> <mi>F</mi><mrow><mo>(</mo> <mrow> <mi>X</mi><mo>+</mo><mi>Y</mi> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>F</mi><mrow><mo>(</mo> <mi>X</mi> <mo>)</mo></mrow><mo>+</mo><mi>F</mi><mrow><mo>(</mo> <mi>Y</mi> <mo>)</mo></mrow><mo>,</mo><mtext>&#x2003;</mtext><mi>F</mi><mrow><mo>(</mo> <mrow> <mi>t</mi><mi>X</mi> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>t</mi><mi>F</mi><mrow><mo>(</mo> <mi>X</mi> <mo>)</mo></mrow> </mrow> </math>$$F\left( {X + Y} \right) = F\left( X \right) + F\left( Y \right),\quad F\left( {tX} \right) = tF\left( X \right)$$ for all X, Y in V and all scalars t. (V and W can be either real or complex vector spaces, but both of the same type.)