# [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!

$$a$$ 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.