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How do i show that $$\\text{var}(ax+b)=a^2\\text{var}(x).$$ since i am reading statistics for the first time, i don't have any idea how to start Expressing the product ax as a linear combination of the column vectors of a ask question asked 14 years, 5 months ago modified 14 years, 5 months ago I suppose it is a trivial step to say that h not changing the norm of any vector x means that it does not change the maximum norm over all vectors ax, but perhaps not.
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In one of the proofs in class there was given the equality for the dot product I was thinking that if the $n$ columns are linearly. $$\langle ax, ax\rangle = \langle x, a^tax\rangle$$ i don't understand why this.
A linear function fixes the origin, whereas an affine function need not do so
An affine function is the composition of a linear function with a translation, so while the linear part. You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful What's reputation and how do i get.
It sounds like you're interested in why equation problems $ax=b$ are called linear while eigenvalue problems $ax=\lambda\cdot x$ are called nonlinear In Rudin's Principles of Mathematical analysis p. 208 $\|A\|$ is defined as the $\sup$ of all numbers $|Ax|$, where $x$ ranges over all vectors in $\mathbb {R}^n$ with $|x|≤1$. Prove that if the columns of the $m\times n$ matrix $a$ are linearly independent, then $ax=b$ has at most one solution
