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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1answer
58 views

linear combination

If $\mathbb C$ is the field of complex numbers which vectors in $\mathbb C^3$ are linear combinations of $(1,0,-1)$,$(0,1,1)$ and$(1,1,1)$? Please help.
6
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3answers
234 views

Learning Math At Home

I want to learn math on my own at home. What is the best method to do so? I would say that I pick things up/grasp concepts pretty fast. I took math until grade 10 in highschool/secondary school and ...
3
votes
1answer
97 views

Eigenvalues and Eigenvectors Diagonilization

Let $ A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$. Ok so the first thing I need to look ...
0
votes
1answer
135 views

Nullspace of Vandermonde matrix

Let ${\bf A} \in\mathbb{R}^{M\times N}$ Vandermonde matrix, $N>M$ $$A=\left(\begin{array}{ccc} 1 & 1 & \dots & 1 \\ \alpha_1 & \alpha_{2} & \dots & \alpha_{N} \\ \vdots ...
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votes
0answers
22 views

How to know the +,- distance made by a point which is not exactly on a given plane?

If I have a vertical plane, and if there are few points bit away from the plane (nearly 5 to 10 cm), then how to know the + or - distance that point is making with the given plane. Any ...
1
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1answer
245 views

Inner product for a finite dimensional vector space

Do we always have an inner product for a finite dimensional vector space $V$ over a field $k$ such that $V$ is a Hilbert space? Thank you very much.
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2answers
48 views

question on linear algebra-vector spaces-basis

Let $V$ be the space of $2\times2$ matrices over $F$. Find a basis $\{A_1,A_2,A_3,A_4\}$ for $V$ such that $A_j^2=A_j$ for each $j$? Please solve this question I cannot solve it.
0
votes
1answer
75 views

Vector spaces-span

Let $W$ be the set of all $x=(x_1,x_2,x_3,x_4,x_5)$ in $\mathbb{R}^5$ which satisfy $2x_1-x_2+\frac{4}{3}x_3-x_4=0$ $x_1+\frac{2}{3}x_3-x_5=0$ $9x_1-3x_2+6x_3-3x_4-3x_5=0$. Find a finite set of ...
1
vote
5answers
101 views

Can I treat vectors in $\Bbb R^3$ with a component equal to zero as vectors in $\Bbb R^2$?

If I have a vector $(0,4,4)$ and was finding perpendicular unit vectors to this, is it the same case as finding perpendicular unit vectors for the vector $(4,4)$? Meaning a maximum of two possible ...
2
votes
2answers
91 views

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
3
votes
1answer
47 views

bound on $l_2$ error in approximating a vector with its $t$-sparse representation

How do I prove that for any vector $y\in \mathbb{R}^n$, and any positive integer $t$, \begin{equation} ||y-y_t||_2\:\leq\: \frac{1}{2\sqrt{t}}||y||_1 \end{equation} where $y_t\in\mathbb{R}^n$ is the ...
5
votes
1answer
168 views

Cross Product Intuition

I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
0
votes
1answer
15 views

Collection of $5$ vectors in $\mathbb{R}^3$ such that any $3$ are Linearly Independent

I am looking for a collection of $5$ vectors in $\mathbb{R}^3$ such that any $3$ are linearly independent. What I tried to do at first is set up a collection of linear equations like this: $$c_1M_1 + ...
1
vote
1answer
47 views

Question on orthogonal subspaces

I'm given a problem with the initial values $u =\begin{bmatrix}1\\1\\0\end{bmatrix} , v = \begin{bmatrix}2\\3\\0\end{bmatrix}, b =\begin{bmatrix}4\\5\\6\end{bmatrix}$. I've calculated that the ...
-3
votes
2answers
149 views

Why cross product's formulas defined in this way? [closed]

Why cross product's formulas defined in this way? When mathematicians need to define cross product?
2
votes
1answer
96 views

Two quick eigenvalues & complex numbers questions

A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ? ($Im(v)$denoting the imaginary part of the vector $v$) My understanding: since every row of the vector is a complex number (say ...
2
votes
2answers
155 views

$\mbox{Ker} \;S$ is T-invariant, when $TS=ST$

Let $T,S:V\to S$ linear transformations, s.t: $TS=ST$, then $\ker(S)$ is $T$-invariant. My solution: $$\{T(v)\in V:TS(v)=0 \}=\{T(v)\in V:ST(v)=0 \}\subseteq\ker(S)$$ If its right, then why ...
2
votes
3answers
70 views

Densities of planes

I have misunderstanding about what does mean density of planes. For example I was trying to figure out definition of co-vector, and while browsing in internet definition of it, I found ...
1
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1answer
67 views

Number of Subspaces of a Vector Space

If $V$ is an $n(<\infty)$ dimensional vector space over a finite prime field $\mathbb{F}_p$, then it is well known that the number of $1$-dimensional subspaces of $V$ is equal to the number of ...
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1answer
159 views

practice questions - vectors

I'm working my way over some practice questions and have hit two that I have no idea on.  14. Find a vector parametric equation for the plane containing the points whose position vectors ...
0
votes
1answer
65 views

Find all nonzero values of $k$ such that the vectors $(-11; k; 2)$ and $(k; k^2; k)$ are orthogonal.

Find all nonzero values of $k$ such that the vectors $(-11; k; 2)$ and $(k; k^2; k)$ are orthogonal. Can someone help me with this thanks?
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1answer
1k views

Find the vector parametric equation of the line through $A$ and $B$.

Let $A = (1; 2; 1)$ and $B = (2;-1;-1)$. Find the vector parametric equation of the line through $A$ and $B$. If $C = (3;-4;-3)$, show using your equation that $A, B$ and $C$ are collinear. I missed ...
11
votes
3answers
315 views

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
1
vote
5answers
380 views

Linear Transformation from $ \mathbb R^2 \rightarrow \mathbb R^2 $

Let $ v_1 = \begin{bmatrix} 1 \\ -1 \\ \end{bmatrix} $ and $ v_2 = \begin{bmatrix} 2 \\ -3 \\ \end{bmatrix} $ Let $ \mathbb R^2 \rightarrow \mathbb R^2 $ be linear transformation satisfying $ ...
0
votes
1answer
113 views

Problem related to differential of a map

I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
3
votes
1answer
137 views

Formula relating traces of a linear map, its restriction to an invariant subspace and the induced quotient map?

Let $V$ be a finite-dimensional vector space, $T\colon V \to V$ a linear map and $W \subset V$ a $T$-invariant subspace (i.e., $T(W) \subset W$). Then there is a well-defined induced quotient map ...
5
votes
4answers
442 views

Linear Transformations $ \mathbb R^2 \rightarrow \mathbb R^3 $

If $ T : \mathbb R^2 \rightarrow \mathbb R^3 $ is a linear transformation such that $ T \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 12 \\ -2 \end{bmatrix} $ and $ ...
0
votes
1answer
57 views

Basis of the orthogonal component

Let $\{v_1,\dots,v_n\}$ be an orthogonal basis for $R^n$ and let $W=\operatorname{span}\{v_1,\dots,v_k\}$. Is it necessarily true that $W^\perp=\operatorname{span}\{v_{k+1},\dots,v_n\}$? Either prove ...
0
votes
1answer
247 views

Column space of a $QR$-factorization

Let A be an $m$ x $n$ matrix with linearly independent columns and let $A=QR$ be a $QR$-factorization of $A$. Show that $A$ and $Q$ have the same column space. I honestly don't have a clue where to ...
1
vote
1answer
224 views

Orthogonality of vectors and the orthogonal complement

Let $W$ be a subspace of $R^n$ and $v$ a vector in $R^n$. Suppose that $w$ and $w'$ are orthogonal vectors with $w$ in $W$ and that $v$ = $w$ + $w'$. Is it necessarily true that $w'$ is in ...
1
vote
2answers
47 views

linear algebra -vector spaces

If $V$ is a vector space over the field $F$ then verify that $$(\alpha_1+ \alpha_2)+(\alpha_3+\alpha_4)=[\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4$$ for all the vectors ...
2
votes
1answer
89 views

Vector orthogonal projection what's my error

I've been recalculating the following problem for hours and I don't understand what I'm doing wrong, please help me. We have the following figure: By using orthogonal projection, ...
3
votes
1answer
155 views

Finding the dimensions of subspaces of a Vector space and S-cyclic subspaces using minimal poynomials

I've been staring at a chapter in Bill Cooperstein's Advanced Linear ALgebra for some time now and one section is giving me trouble. It is about elementary divisors and invariant factors. My ...
2
votes
1answer
43 views

How to estimate orientation errors of an image with respect to known data (line features)

I think this is very simple but for me, it is confusing to figure out a way. Here is my problem. I have been given a 3d line segment list obtained from a field survey. So I know each end point ...
2
votes
1answer
2k views

How does one go about proving that something is a vector space?

So I have this pretty theoretical problem for homework that says I need to show that this set of matrices is a vector space. It says that we have the set $M_{m,n}(\mathbb R)$, $m\times n$ matrices ...
0
votes
3answers
123 views

Basis of the subspace of $\mathbb R^4$

Find a basis of the subspace of R4 consisiting of all vectors of the form: $$\begin{bmatrix}x_1\\ 6 x_1 + x_2\\ 4 x_1 + 5 x_2\\ 8 x_1 - 9 x_2\end{bmatrix}$$ Now, I really have no clue how to set ...
1
vote
1answer
51 views

How to estimate (if/any) displacement/rotations between 2d line segments taken from 2 data sets

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I derive these two line sets using image based (e.g. CD) and manual method ...
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votes
4answers
405 views

Proofs: Subspaces, vectors.

Studying for an exam, came across this question: Let $V$ be a subspace of $\mathbb{R}^n$ and $u \in \mathbb{R}^n$ but $u \notin V$. $W = \{v + cu: v$ is in $V$ and $c$ is a scalar$\}$ A) Prove ...
0
votes
1answer
43 views

How to fit an object of constant size based on measurements to known points

I'm looking for a mathematical solution for solving where the base of a camera crane (ie a constant square or rectangle of known dimensions) is with measurements to known points. This seems to be a ...
1
vote
1answer
95 views

angle between steepest gradient of two plane

IF I have two 3d planes such as Oab and Oa'b'. If these two planes intersect a horizontal plane and the intersection of each plane makes AB and A'B' lines. then, Does the angle between AB, A'B' ...
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votes
1answer
65 views

$2$ planes and angle between them

IF I have two $3d$ planes such as Oab and Oa'b'. If these two planes intersect a horizontal plane and the intersection of each plane makes AB and A'B' lines. then, Does the angle between AB, ...
2
votes
1answer
72 views

Proving Bounded variation is smallest linear space

Prove that $BV[a,b]$ is the smallest linear space containing all monotone functions on $[a,b].$
3
votes
1answer
72 views

Some questions about quaternions.

It is possible make something like complexification of a real vector space using quaternions? If yes, it's similar to complex case or there are considerable differences? Has been studied a quaternion ...
0
votes
1answer
76 views

Find a number that minimizes distance to a vector of sets of numbers

Assumptions $V$ is a vector of sets $V_1,V_2,...,V_n$ of numbers: $V=[V_1, V_2,..., V_n]^T, \forall_{i=1..n}V_i\subset\mathbb{R}$ $c\in\mathbb{R}$ is constant $d(V,c)$ is an error metric: ...
0
votes
2answers
182 views

what is $ M^{\perp}$ given set?

Let $ ‎X=C[-1,1]‎$‎‎ be inner product space with definition $$‎\langle f,g‎‎‎\rangle =‎\int_{-1}^1 f‎‎ \overline{g}‎ ‎dt ‎‎.$$ Let $M$ be the subspace defined by ‎$$ ‎M= ‎‎\left\{f‎ \in ‎X\mid ...
0
votes
1answer
301 views

Finding a vector in the plane of one vector and orthogonal to another

Given vectors a, b, and p, I am trying to find unit vector u, such that u is in the plane spanned by a and p and is orthogonal to b. Assume all vectors are unit vectors I can form 3 equations: u.b ...
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3answers
160 views

Is this set a Vector Space or not?

Is the set of matrices of the form $\begin{pmatrix} 1 &a\\ b&1 \end{pmatrix}$ a vector space? Answer is no. Let me try; In order to be a vector space, the following two criteria must be ...
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2answers
63 views

Do you have any idea what matrix space is?

Can someone explain what matrix space is by a given example. Is it a set of number?
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votes
1answer
106 views

Steepest slope gradient of a vertical plane

I know the steepest slope gradient (Azimuth) of a 3D plane can be obtained by projecting normal vector onto XY Plane. So, when the plane is slant, the steepest gradient will be a some value. ...
2
votes
2answers
62 views

Vector Dimensionality

I'm sure this is a pretty basic concept, but I'd like to clarify the following. Assuming $\vec{v} = (v_{1}, \ldots, v_{n})$, for $n > 1$; naturally, $\vec{v}$ has a "presence" (or coordinates) in ...