Thread: Lin. Transformation

Let T be the Lin. Trans. $\displaystyle p_2 \rightarrow p_2$ defined by

$\displaystyle T(at^2+bt+c) = (3a+2c)t + (2b-5c)$

1.) Determine a basis of the kernel of $\displaystyle T$

2.) Determine a basis of the range of $\displaystyle T$

3.) Is $\displaystyle T$ onto? Is $\displaystyle T$ one-to-one? Explain!


....

Well for #3 I know one-to-one has to do w/ range and onto has to do w/ kernel...

Some help?

Originally Posted by caeder012

Let T be the Lin. Trans. $\displaystyle p_2 \rightarrow p_2$ defined by

$\displaystyle T(at^2+bt+c) = (3a+2c)t + (2b-5c)$

1.) Determine a basis of the kernel of $\displaystyle T$

2.) Determine a basis of the range of $\displaystyle T$

3.) Is $\displaystyle T$ onto? Is $\displaystyle T$ one-to-one? Explain!


....

Well for #3 I know one-to-one has to do w/ range and onto has to do w/ kernel...

Some help?

Is the basis of the kernel basis = span{$\displaystyle -4t^2 + 15t + 6$}

because those coefficients make $\displaystyle (3a+2c) + (2b- 5c) = 0$

And for range.. range = span {$\displaystyle t + 1$}

Hmm..

Originally Posted by caeder012

Is the basis of the kernel basis = span{$\displaystyle -4t^2 + 15t + 6$}

because those coefficients make $\displaystyle (3a+2c) + (2b- 5c) = 0$

And for range.. range = span {$\displaystyle t + 1$}

Hmm..

Anyone have some insight? One of the hardest lin alg problems I've come across.

Shouldn't you be thinking about:

3a + 2c = 0

AND

2b - 5c = 0

????

You certainly found one solution to this over-defined system. What about the infinitely many others?

May 'a' be zero (0)?