Linear Algebra: UNIT III: Inner Product Spaces
Inner Products and Norms: Example Solved Problems - Part 1
Important Example Solved Problems - Engineering Maths or Mathematics - Inner Products and Norms: Example Solved Problems - Part 1
INNER PRODUCTS AND
NORMS
Example Solved Problems
– Part 1
Example 1
Let U = (2,−3, 6) &
V= (8,2,−3). Find U.V
Solution:
U.V (16−6−18) = −8.
U.V exists.
Suppose
that U= (1, −3, 0, 5) & V= (3, 6, 4).
Here
U.V doesn't exist. Because U is an 1×4 and V is an 1×3 structures.
Example 2
Let A = 
. Find
the conjugate transpose.
Solution:
Example 3
Let V=Mn×n(F)
and define <A, B> = trace (B*A) for A, B ∈
V. Verify it is an inner product or not.
Solution:
Let
A, B, C ∈ V & V=Mn×n(F)
To
prove it is an inner product it should be satisfied the 4 condition
(i)
<x+z, y> = <x, y> + <z, y>
(ie) <A+B, C> = tr (c* (A+B))
= tr (c* A+c* B) = <A, c> + <B, c>
(ii)
<cx, y> = c<x, y>
(ie)
<kA, B> = tr (B* kA) = tr (kB* A) = k<A, B>.
(iii) 
= <y, x> ⇒ 
= <B, A>
(iv)
<x, x> > 0 if x≠0.
(ie) <A, A> = tr (A* A) =
If A ≠ 0, then Aki ≠
0 for some k and i. So <A,A> > 0.
This
inner product on Mn×n(F) is called Frobenius inner product.
Example 4
Prove that:
Solution:
Let
the standard inner products be
Linear Algebra: UNIT III: Inner Product Spaces : Tag: maths, mathematics : - Inner Products and Norms: Example Solved Problems - Part 1
Linear Algebra: UNIT III: Inner Product Spaces
Under Subject
Linear Algebra
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