Question

2. The Euclidean norm (2-norm) of an real N-dimensional vector x is its length (the distance from the origin to the point x in N-dimensional hyperspace $\mathbb{R}^N$). It is calculated as $\|\|x\|\| = \sqrt{\sum_{n=1}^{N} x^2(n)}$, $<x, y> = \sum_{n=1}^{N} x(n)y(n)$ (a) Rewrite $<x, y>$ in terms of matrix-vector operations. (b) Define $\|\|x\|\|$ in terms of the inner product $<...>$ (c) Simplify $<ax, y>$, where a is a scalar constant. (d) Using the inner product $<...>, rewrite the MSE expression $E = \frac{1}{N_v} \sum_{p=1}^{N_v} \sum_{i=1}^{M} [t_p(i) - y_p(i)]^2$ twice, once using $t_p(i)$ and $y_p(i)$ which have dimensions $N_v$ by 1 and again using $t_p$ and $y_p$ which have dimensions M by 1.

          2. The Euclidean norm (2-norm) of an real N-dimensional vector x is its length (the distance
from the origin to the point x in N-dimensional hyperspace $\mathbb{R}^N$). It is calculated as
$\|\|x\|\| = \sqrt{\sum_{n=1}^{N} x^2(n)}$, $<x, y> = \sum_{n=1}^{N} x(n)y(n)$
(a) Rewrite $<x, y>$ in terms of matrix-vector operations.
(b) Define $\|\|x\|\|$ in terms of the inner product $<...>$ 
(c) Simplify $<ax, y>$, where a is a scalar constant.
(d) Using the inner product $<...>, rewrite the MSE expression
$E = \frac{1}{N_v} \sum_{p=1}^{N_v} \sum_{i=1}^{M} [t_p(i) - y_p(i)]^2$
twice, once using $t_p(i)$ and $y_p(i)$ which have dimensions $N_v$ by 1 and again using $t_p$ and $y_p$
which have dimensions M by 1.

2. The Euclidean norm (2-norm) of an real N-dimensional vector x is its length (the distance
from the origin to the point x in N-dimensional hyperspace ℝ^N). It is calculated as
x = √(∑n=1^N x^2(n)), <x, y> = ∑n=1^N x(n)y(n)
(a) Rewrite <x, y> in terms of matrix-vector operations.
(b) Define x in terms of the inner product <...>
(c) Simplify <ax, y>, where a is a scalar constant.
(d) Using the inner product <...>, rewrite the MSE expressionE = (1)/(Nv) ∑p=1^Nv ∑i=1^M [tp(i) - yp(i)]^2twice, once usingtp(i)andyp(i)which have dimensionsNvby 1 and again usingtpandypwhich have dimensions M by 1.

Added by Robert M.

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