All Questions
Tagged with vector-spaces inner-products
650
questions
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
- 1
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0
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23
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Graphical Intuition of a Linear Transformation in terms of Row Vectors
The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me:
...
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103
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Why is the inner product space defined separately?
While learning about the inner product space, I became curious
why it is defined separately?
In my opinion, there seems to be no difference between defining the inner product space separately and ...
- 19
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45
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Manifolds and Euclidean Spaces
This might be a basic question, but it's been irking me for the past few days.
The common definition of a manifold is as a second-countable, Haussdorff topoplogical space which is locally homeomorphic ...
- 358
2
votes
1
answer
52
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Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.
Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
user1315539
1
vote
1
answer
28
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Given an inconsistent overdeterminate system AX=b where $A\in M_{m×n}(R)$ and $b\in R^m$ with rank A=n. Find the least square approx. solution of AX=b
Suppose $A$ is a real matrix of order $m\times n$ with $m>n,b\in\Bbb R^m$ be such that the over determined system
of linear equations $AX=b$ is inconsistent and $\text{rank} (A)= n.$ Let $W$ be the ...
- 3,073
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2
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56
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$\lambda \in \mathbb{C}$ is an eigenvalue of the operator $A$, then $\text{Re}(\lambda) = 0$ AND $H$ is a complex vector space, then $A = iB$.
Let $H$ be a Hilbert space and $A \in B(H)$ such that $A^* = -A$. Prove the following statements:
(a) If $H$ is a real vector space, then $\langle Ax, x \rangle = 0$ for every $x \in H$.
(b) If $\...
user1316790
0
votes
1
answer
65
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True or False: Inner product on $\mathbb{R}^2$ satisfying a specific norm.
Verify or refute: There exists an inner product in $\mathbb{R}^2$ such that the norm of every vector $v=(v_1,v_2)$ is $\|v\|=|v_1|+|v_2|$.
I think this is untrue. So I took $v=(1,0), y=(0,1)$. After ...
- 1,494
1
vote
1
answer
74
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Orthogonal projection is bounded
Definition: Let $U$ be a subspace of $V$. The orthogonal projection of $V$ onto $U$ is the operator $P_U\in L(V)$ given by
$$P_U(u+w)=u$$ if $u\in U, w\in U^{\perp}$.
Let $V$ be a space with inner ...
- 1,494
2
votes
1
answer
52
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Consequences of definition of scalar product
Definition: Let $V$ be a vector space over the field $K=\mathbb{R}$ (or over $K=\mathbb{C})$. The scalar product on $V$ is a function $V\times V\to K,$ denoted by $(x,y)\mapsto \langle x,y\rangle$, ...
- 1,494
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3
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104
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Why is $ R(A^*) \perp N(A)$ true?
Let a matrix the $A \in M_{n\times n}(\mathbb{C})$. My question is:
(1) Why every matrix $A$ satisfies
$ R(A^*) \perp N(A)$(where $R(A),N(A)$ are range of $A$,null space of $A$ respectively)?
And why ...
user1290851
0
votes
1
answer
38
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Projection of vector
The projection of a vector $x$
onto a vector $u$ is
$proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$
Projection onto $u$
is given by matrix multiplication
$proj_u(x)=Px$ where $P=\frac{...
user1290851
1
vote
1
answer
52
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The inner product of column vectors is the same as matrix multiplication
I am very much new on the topic of inner product:
Definition. The inner product of vectors $x, y \in \mathbb{R}^n$
is $\langle x, y\rangle =\sum_{i=1}^{n} x_ky_k=x_1y_1+x_2y_2+\dots+x_ny_n$
I can't ...
user1290851
2
votes
1
answer
38
views
Matrix of an Inner Product and Spectral Theorem
My linear algebra has become very rusty and now I've confused myself entirely.
Let $V$ be an inner product space over an $n$-dimensional real vector space $V$. Moreover, let the set of vectors $$\...
- 515
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1
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59
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Orthogonal orthornomal bases imply pair-orthogonal vectors
While self-studying linear algebra i started thinking about following problem: Let's say that $A, B \in \mathbb{C}_{n\times n}$ are orthogonal in a Frobenius sense orthonormal bases of complex vector ...
