# Tagged Questions

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### Showing equivalence of the positivity condition of inner products

I have no idea how to prove this, first off, because I don't think I understand the question. Isn't the second case not true for v = 0. Show that for real vectors spaces $V$ with $V$ $\not= {0}$, ...
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### Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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### Matrix Composed of Traces of Linear Independent Set of Matrices

I got the following problem: Let $S=\{A_1,A_2,...,A_k\} \subseteq \mathbb{M^R}_{n\times n}$ be a linear independent set of $k$ real $n \times n$ matrices with respect to the standard matrix inner ...
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### What am I doing wrong? Gram Schmidt process..

Let there be the inner product of all polynomials of degree smaller or equal to 2: $\langle f,g\rangle=\int_0^1f(x)g(x)xdx$. Find orthonormal basis. So I really tried this for an hour and it pretty ...
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### Does this function define an inner product?

Does the function below define an inner product? $$\langle (x, y), (z, t)\rangle = xz − yt$$ I know how to prove it given two vectors (e.g. $\langle(x,y),(z,t)\rangle$) demonstrating symmetry, ...
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### Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
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### Orthogonal complement of a single vector

Is there a quick way to show that if v is an element of an n dimensional real inner product space space, the orthogonal complement of v is n-1 dimensional? I can do this by using gram-schmidt and ...
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### Proof that two spans are equal

I have an orthogonal subset of nonzero vectors $u_1,\dots,u_n$. I take $v \in V$ (a vector space) so that $v$ is the in the span of the previous set. Now I let $u$ be $v-m_1u_1-\dots-m_nu_n$ with ...
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### Question about definite positive symmetric bilinear form

Let $A$ be the matrix of a positive definite symmetric bilinear form. Prove $a_{11}a_{nn}\ge a_{1n}a_{n1}$. I don't really have a clue of how to solve this.
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### Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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### What does it mean for a vector space to 'have' a particular inner product?

If a vector space 'has' a particular inner product, what does that mean exactly? Is it that all the vectors in the space satisfy the conditions for the inner product to work? Is it that all the ...
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### General Proof that $\left< v, v \right> = \left|\left| v \right|\right|^2$

Consider $\mathbb{R}^n$ with the standard Euclidean inner-product. I'm trying to give a proof that $$\left< v, v \right> = \left|\left| v \right|\right|^2$$ but can only seem to do it for ...
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### Must unitary matrices satisfying this property commute?

If A and B are unitary matrices such that A, B, and AB are all conjugate to diag(1,1,-1,-1), must AB=BA? Why or why not?
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### Adjoints and Inner Product Spaces

The proposition states that if $V$ is a finite dimensional IPS and $\ \phi : V \rightarrow V$ is a linear operator, then $\phi$ has an adjoint $\phi^*$ and if the matrix of $\phi$ wrt to an ...
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### Prove that [B]ij = <ui, uj>

Let B = A*A. That is, the product of a matrix and its transpose. Prove that the $[B]_{ij} = < u_{i}, u_{j}>$. Now, upon looking at this, I decide to test it out first to see if I believe it. I ...
6. Consider $V = \def\P{\mathbb P^2}\P$ with inner product: $$\def\sp#1{\left<#1\right>}\sp{p(X), q(X)} = 2p(-1)q(-1) + 3p(1)q(1) + p(2)q(2)$$ a. Show that for any non-zero polynomial ...
I'm having trouble with this excercise: Determine the orthogonal space of odd continuous functions in the space of continuous functions. (In the interval $[-1,1]$) I assume that the inner product ...