[Math] Inner Product with a Linear Transformation

inner-productslinear algebralinear-transformations

$W$ is a vector space equipped with the inner product $\langle\;,\;\rangle$. T is a linear transformation from W to W. Under which condition(s) does the map $\langle a, b\rangle_1$ $=$ $\langle T(a), T(b)\rangle$ define an inner product on $W$?

I thought that this map would define an inner product if and only if either $a$ or $b$ were the zero vector, or if both were the zero vector. This was because I felt that the only way for this linear map and linear transformation to be positive definite was if the zero vector transformed itself to the zero vector. But I do feel that this map is symmetric and bilinear.

However, since the transformation itself does not yield a specific value, I am a little confused on how I would identify what the number that is output is if we are not solely focusing on the vectors $a$ and $b$. Because to check for positive definiteness, we need to verify that the only way for, say, $\langle T(a), T(a)\rangle $ to be greater than or equal to zero is if $a$ is the zero vector.

Any assistance would be very much appreciated!

Best Answer

$a$ and $b$ are variable vectors and you have to find condition(s) on $T$ (not on these vectors) which make(s) $\langle a,b\rangle_1$ an inner product. Apply definition of inner product. $\langle a,b\rangle_1$ satisfies all properties of inner product for any $T$ except the condition $\langle a,a\rangle_1=0$ implies $a=0$. In other words what we need is $\|Tx\|=0$ implies $x=0$. This is true iff $T$ is injective.

Related Question