# Tagged Questions

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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### Finding The Shortest Distance Between Two 3D Line Segments

I have two Line Segments, represented by a 3D point at their beginning/end points. Line: ...
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### Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
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### Proving set of upper triangular matrices form subspace of $M_{m \times n}$ (F)

Proving set of upper triangular matrices form subspace of $M_{m \times n}$ (F) Now clearly if i set all elements including diagonal =0 then we can say that o vector is there. Also it satisfies ...
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### satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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### What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular? [on hold]

What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular?
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### Showing that a function in a vector space is linear

Let $X$ be a vector space and consider a function $f : X \rightarrow \mathbb{R}$ defined for some $a \in X$ defined as $f_a (x) = a \cdot x$. (i) Prove that $f_a (x) = a \cdot x$ is a linear function....
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### Why are vectors portrayed as a sub space of $R^n$

Typically I have seen all vector notation showing it as a sub space of $R^n$. Why not $Q$ or $Z$?
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### Let V subspace of W, with dimV=dimW. Why should they be equal?

The fact that $V$ is a subspace of $W$ , means $\dim V \leq \dim W$ . We are told $\dim V = \dim W.$ So if $B_1$ is the basis of $V$ and $B_2$ is the basis of $W,$ it is $|B_1|= |B_2| .$ We know that ...
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### Is the finite dimension of a vector space over the complex numbers half the dimension of the same vector space considered over the reals?

Consider a vector space V with basis ${b_{1},..,b_{n}}$ and complex scalars. This obviously has dimension n. Now consider a space with the same exact set of vectors of V, except with real scalars. I ...
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### Two vector spaces with same dimension and same basis, are identical?

Let $V$ subspace of $W$ and both have same dimension and same basis. Then can we safely say that $V= W$ ? I believe yes. For example there may be an element $x \in V$ written as a linear combination ...
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### Find m so that $(m+1,1,1)$ , $(1,-m,-1)$ , $(m,1-m,2)$ are linearly dependent

I formed an augmented matrix $$\left(\begin{array}{ccc|c}m+1&1&m&0\\1&-m&1-m&0\\1&-1&2&0\end{array}\right)$$ I now that we do reduced row echelon form for the ...
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### Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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### How to prove $A^{\perp}$ is a closed linear subspace?

Suppose $X$ is an inner product space and $A\subseteq X$. I need to prove that $A^{\perp}$ is a closed linear subspace of $X$. Can anyone give me a idea?
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### Is there a linear transformation that sends v to (1,0)

Suppose we are given a linear vector field $v(x) = Ax$ on $\Bbb R^2$. Suppose $x_0$ is the point where this vector field does not vanish. Is it possible to find a linear transformation from $\Bbb R^2$ ...
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### Space spanned by matrices

I have a set of $5 \times 5$ matrices, $M_1, M_2,\dots, M_{19}, M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I ...
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### Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
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### $V$ be a vector space , $T:V \to V$ be a linear operator , then is $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$? [duplicate]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$ ? (note that the direct product is well-defined as both the spaces ...
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### Find the vector equation of a line passing through point A perpendicular to line AB

'Points A and B have coordinates (4,1) and (2,-5) respectively. Find a vector equation for the line which passes through the point A (only the point A), and which is perpendicular to the line AB.' ...
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### Dimension of kernel for nilpotent transformation powers

Let $T:\Bbb R^n \to \Bbb R^n$ be a nilpotent transformation with index $n$. (i.e. $T^n=0$). Is it true that for all $n≥k≥0$, $\dim \ker T^k=k$? How can that be shown? The context is a linear algebra ...
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### Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: h = \sum_n \langle h,h_n\...
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### $V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2)$?

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2)$ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement ...
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### Intuitive way to understand the use of matrix inversion to find dual basis

I'm currently thinking about the following problem: Problem: Let $B = (b_1, b_2, b_3)$ a base of $\mathbb{R}^3$. Find the correlating dual basis $B^* = (b_1^*, b_2^*, b_3^*)$. $B$ is explicitly ...
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### What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
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### linearly dependent family of vectors.

Can someone help me to solve this question please : Establish, by induction, that : $\forall n \in \mathbb{N} \setminus \{ 0,1 \} \ \forall v_1 , \dots , v_n \in \mathbb{R}^n$ linearly independants ...
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### 'Tetrahedral' coordinates in space (generalization of hexagonal coordinates)

The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal. Imagine ...
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### Linearly Independent set of vectors that spans the same subspace of $\mathbb{R}^3$

I'm having trouble setting this up. I have these $3$ column vectors: $\langle 1, 1, 2\rangle$ $\langle -7, -1, -8\rangle$ $\langle 3, 0, 3\rangle$ I need to find a linearly independent set of ...
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### Why are not these two sets subspaces of $\mathbb{R}^3$?

Why are not these two sets subspaces of $\mathbb{R}^3$?  \begin{align} S_1&=\left\{\begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}:x_1=x_3\text{ or }x_2=-2x_3 \right\}\\ S_2&=\left\{\begin{...
Say I have a space $V^{(1)}$ with basis $\{a_i \}$ and $V^{(2)}$ (with dimensions $d_1$, $d_2$ respectively) with basis $\{b_j\}$. Clearly the vectors $\{a_i\otimes b_j\}$ are a basis for \$V^{(1)}\...