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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Is $\mathrm{span}\{v_1,v_2,v_3\}=\mathrm{span}\{v_1+v_2,v_1+v_3,v_2+v_3\}$ if $v_1,v_2,v_3\in V$ a vector space over $\mathbb{Z}_2$?

Is $\mathrm{span}\{v_1,v_2,v_3\}=\mathrm{span}\{v_1+v_2,v_1+v_3,v_2+v_3\}$ if $v_1,v_2,v_3 \in V$ a vector space over $\mathbb{Z}_2$? I know that this is true if you assume that $V$ is over any ...
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1answer
56 views

Term for “pre-image, but in a vector-space-y way”?

Consider a linear transformation $T :: V \rightarrow W$. Today I found myself wanting to use "the pre-image of $T$" to talk about "the subspace of $V$ that doesn't get killed by $T$". I guess that ...
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Show that the column space of a matrix is not equal to $\mathbb R^3$

Show that the column space of \begin{pmatrix} 4 &−1& 2 \\ 0 &0& 0 \\ 5 &−1 &6 \\ \end{pmatrix} is not equal to $\mathbb R^3$. I have begun by setting my vector ...
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1answer
43 views

Linear Algebra: showing that $\langle x,y\rangle$ is an inner product on Rn.

Just a simple example: Let $V=\mathbb R^n$, Define $\langle x,y\rangle = x^Ty = x_1y_1+...+x_ny_n$. Verify that the function $\langle x,y\rangle $ satisfies the four conditions of being an inner ...
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1answer
32 views

Prove that $V = V_1 \oplus V_2$ in the following

Let $V$ be a vector space over $\mathbb{C}$ such that $T^2=1_V$. Define $V_1 = \left\{v\in V |\ T(v) = v \right\}$, $\ V_2 = \left\{v\in V |\ T(v) = -v\right\}$ prove that $V = V_1 \oplus V_2$ For ...
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2answers
50 views

Vector Space, Dimension, Subspaces

Suppose that V is a vector space and dim(V) = 4. W is a subspace of V. Prove directly that W must have finite dimension. My Answer: Since W is a subspace of V, dim(W) must be less or equal to ...
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1answer
19 views

Basis & Dimension for Joint Subspaces

Assumption: Assume that $V_1$ and $V_2$ are subspaces of $\mathbb{R}^\mathbb{3}$ Question: "Suppose that $V_1$ is the subspace of $\mathbb{R}^\mathbb{3}$ given by $V_1 = \{(2t-s, 3t, t+2s): t, ...
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1answer
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Prove that dim(nullspace($ST$)) = dim(nullspace($T$)) + dim(range($T$) ∩ nullspace($S$)).

Suppose $U$ is a finite-dimensional vector spaces, that $S$ ∈ $L(V,W)$, and that $T$ ∈ $L(U, V )$. Prove that dim(nullspace($ST$)) = dim(nullspace($T$)) + dim(range($T$) ∩ nullspace($S$)). $S\circ ...
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Finding a subspace using combination formula

This is a question from my textbook The answer I have is something like this: let $X_1$ and $X_2$ $\in K$,and a,b $\in R$, because $T(aX_1+bX_2)=aT(X_1)+bT(X_2)=0+0=0$, so K is a subspace of ...
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1answer
48 views

Linear Algebra, Vector Spaces (Subspace Theorem)

Question: "Let F(−∞, ∞) represents the set of all real valued function defined on (−∞, ∞). Using the subspace theorem, show that the set of all differentiable functions on (−∞, ∞) that satisfy ...
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1answer
18 views

Find the dimension of the nullspace and basis of the following $T$

Let $V$ be the vector space of polynomials of degree at most 999 with real coefficients. Define a linear map: $T:V\rightarrow\mathbb{R^{100}}\ $ where $T(p) = (p(1),p(2),......,p(100))\ $ ($p$ ...
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1answer
44 views

Subset, Dimension and Vector Space

Suppose that $\{v_1,v_2,v_3\}$ is a linearly independent subset of a vector space $V$, with $\dim(V) = 4$, and that $v_4 \not\in \text{span}\{v_1,v_2,v_3\}$. Prove that $\{v_1,v_2,v_3,v_4\}$ is a ...
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4answers
46 views

What is a simple means of proving that 3 vectors belonging to $\Bbb{R}^2$ are linearly dependent?

For my linear algebra class, there is a 2 part problem that asks, given the set {(1 2), (-1 -1), (1 0)}, prove or disprove that it is linearly independent using the definition only AND then prove or ...
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0answers
23 views

Characteristic Polynomial of a Linear Transformation Proof

Let $T: V\to V$ be a linear transformation of the $n$-dimensional vector space $V$, and suppose there is $v\in V$ such that the set $B=\{v, Tv, T^2 v,\ldots, T^{n-1} v\}$ is a basis of $V$. Let $f$ be ...
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2answers
127 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
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1answer
20 views

Vector spaces: understanding a step towards canonical forms

I am studying canonical forms of matrices, and I'm a little stuck on something: Consider a finite dimensional $\mathbb{F}-$vector space $V$ and an endomorphism $\alpha:\;V\to V$, and let $V_{\alpha}$ ...
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1answer
19 views

Is this a hyperplane or a half-space in $\mathbb{F}_2^n$?

Simple terminological question: the equation $x_1+\dots+x_n = 0$ over $\mathbb{F}_2^n$ is called a subspace. I'm wondering if we could also call it a hyperplane, a half-space or neither? The equality ...
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1answer
79 views

Extreme points of intersection of the orthant (quadrant) with an Hyperplane in finite dimension vectorial space

Note : Having spent some time over the original problem below, I saw that it can be boiled down to a simpler problem. Here is that simpler problem : In a vectorial space (over $\mathbb{R}$) of ...
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2answers
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Extending basis for span of 3 vectors into basis for $R^4$

So I have been reading countless posts on extending a matrix basis, however I still am unable to grasp it and apply what I've read to my problem. w1 = [1,0,-2,3] w2 = [1,-2,3,-1] w3 = [1,-8,4,0] W = ...
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1answer
52 views

How to tell if two vectors span the same space?

Assume we have two vectors with five components. Namely, $v = (v_1, v_2, v_3, v_4, v_5)$ and $u = (u_1, u_2, u_3, u_4, u_5)$. I know that if they are linearly independent, the space that they span ...
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259 views

Finding the Dimension of a given space V

I am unsure how to solve for this problem: If $\vec{v}$ is any nonzero vector in $\mathbb{R}^2$, what is the dimension of the space $V$ of all $2 \times 2$ matrices for which $\vec{v}$ is an ...
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1answer
23 views

Homomorphism of Matrices over Extended Field

Let $E$ and $F$ be fields such that $E$ is an extension field of $F$ and $[E:F] = n$. Let $M_k(F)$ denote the ring of $k \times k$ matrices over $F$. Does there exists a homomorphism from $M_k(E)$ to ...
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1answer
16 views

Is a n multi-step proof rigorous and verification of one.

I have been working through Linear Algebra Done Right by Axler. I came upon this problem and was wondering if my answer was correct. If $U_1,...,U_n$ are finite-dimensional subspace of a vector ...
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0answers
37 views

vector-valued function space definition except for measure zero

I am wondering what's the correct way to mathematically describe the following problem. Say you have an object that can be defined as an open set $\Omega \in \mathbb{R}^d$, where the dimension $ ...
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0answers
15 views

Finding explicit formulas for dual basis.

We're given a vector space $ V = P_2 (R)$ and its ordered basis $\beta = \{1 , x,x^{2}\}$. We need to find the explicit formulas for the vectors of the dual basis $ \beta ^{*}$ for $V^{*}$. Letting $ ...
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A puzzle about vector space

When beginning to study the notion of vector space using the book"linear algebra and its applications" of D.C.Lay,the book give an example of vector space.It says all continuous functions on the ...
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Confused about the definition of subspace

The definition from my textbook is: A subspace of a vector space is a set of vectors that satisfies two requirements: If $v$ and $w$ are vectors in the subspace and $c$ is any scalar, ...
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1answer
46 views

what does linearly independent in C[0, 1] mean?

This is a question from my textbook I'm not quite sure what C[0, 1] mean, I tried to google the similar question and found that $C[0,1]$ usually denotes the collection of continuous functions $f: ...
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1answer
15 views

Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?
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1answer
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how does row vectors span a row space?

the column space is very easy to understand, for example, we have: $$\begin{pmatrix} 1& 0& 0 \\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}\times \begin{pmatrix} a \\ b\\ c ...
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1answer
65 views

Vector as a tensor

If we define a $(p, q)$-tensor $T$ to the vector space $V$ as a multi-linear map: $$ T : \underbrace{V^* \times \dots \times V^*}_{p} \times \underbrace{V \times \dots \times V}_{q} \to \mathbb{R} $$ ...
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How to find $W^{\perp}$ in the following polynomial inner product space?

Consider $P_3(\Bbb{R})$ with inner product $\langle p(x),q(x)\rangle=\int^1_{-1} p(x)q(x)dx$ and let $W=\{ p(x)\in P_3(\Bbb{R})|p(0)=p'(0)=p''(0)=0\}$. How to find $W^{\perp}$? Let's set ...
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2answers
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How to prove that $P_{w^\perp}=I-P_{w}$ for the following condition?

Let $V$ be a vector space over $\Bbb{C}$ with inner product $\langle\cdot,\cdot \rangle$. Let $W$ be a subspace of $V$ and let $P_W:V\to V$ be an orthogonal projection onto $W$ and let ...
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1answer
50 views

Find a basis for a space of functions

Let $X = \lbrace x_1, x_2, . . .,x_n \rbrace$. Find a basis for the space $\mbox{Map}(X,\mathbb R)$. Note: $Map(X,R)$ is the space of all functions that goes from $X$ to $\mathbb R$. Note 2: I was ...
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1answer
17 views

Set of transformations to get a point on the X-Axis.

I have a two points in the 3D coordinate space. Now, I want to send one of the points to the origin and make it (the line joining the two points) align with the X-axis and get the transformation ...
2
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1answer
35 views

How to find the closest point to three vector lines?

So this is the question here I know the angles $A$ and $B$ for each individual, and their positions in longitude and latitude (assuming height of person $z =0$), am I correct in thinking that for any ...
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3answers
13 views

any choice of two columns is a basis for this column space?

This is from my textbook, let's say we have a matirx A and its RREF matrix R $$A=\begin{pmatrix}1 & 0 &3 \\2 & 1 & 5\\1 & 0 & 3 \end{pmatrix}$$ $$R=\begin{pmatrix}1 & 0 ...
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Counting vectors in N-dimensions

Consider an n-dimensional world where vectors can be filled in using $1,-1$ only. We are given a vector $V$ of $n$ dimension. A vector is called valid if it does not contain the same integer at $3$ ...
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Further Readings on Linear Algebra

I am currently working on Linear Algebra Done right by Sheldon Axler. Out of curiosity I am wondering what would be the next material for Linear Algebra after this book?
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Why is the minimal polynomial of linear transformation $T$ the lcm of all elementary divisor

Suppose you have a vector space $V$ over $K$ and you give $V$ a $K[x]$ module structure via a linear operator $T$. Why then is the minimal polynomial of $T$ necessarily the lowest common multiple of ...
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Equivalence of definitions of minimal polynomial of a linear transformation $T$ over a vector space

Let $V$ be a vector space over $K$ and let $T\in End_{K}(V)$. Then $V$ has a $K[x]$-module structure via the operation defined as $f(x)v=f(T(v))$. Using this we can know a lot about the structure of ...
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5answers
862 views

What is the point of subspaces? [duplicate]

I've been studying for my linear algebra final for the past few days, and just had this thought. I know what a subspace is; It's a subset of a vector space that contains the zero vector, and ...
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2answers
31 views

Finding a basis for $\text{span}\{x^{-2} - x, 2 - x^{-1}\}$

I need to find a basis for $V = \text{span}\{x^{-2} - x, 2 - x^{-1}\}$. Clearly $x^{-2} - x$ and $2-x^{-1}$ are linearly independent. So would it suffice to say that a basis for $\text{span}\{x^{-2} - ...
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1answer
22 views

Definition of Outer Product of (abstract) vectors

I was reading an article from the American Mathematical Monthly on the Caratheodory derivative for functions of several variables, and in one of the proofs the authors construct a linear ...
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1answer
34 views

Doesthis argument prove that circle is an $\mathbb R$-vector space?

I'm going to make the unit circle $S^1$ into a vector space. I made the following proof which seems to work well. Just wondering if also you think so. Please let me know if you think there is any flaw ...
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Understanding complex multiplication as vector addition

I've been studying complex multiplication and vector math lately. Below is a visual representation of what happens when one multiplies a complex number $ 1 + xi $ by itself repeatedly, plotting the ...
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1answer
34 views

dimension of vector space constructed by reducing another vector space

given $C \in F^n$ is a vector space of dimension $k$, we construct a new vector space like so: fix an integer $1\le i \le n$ find all $\overrightarrow{c}\in C$ that uphold $c_i=0$ delete that ...
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0answers
85 views

Derivative of Elementwise Function (working on a vector)

I have seen an example (it is in terms of neural network back propagation) that I dont understand. Given: $\textbf{a} = \textbf{x}\textbf{W}_{1}+\textbf{b}_{1} $ (where x is dimension (1x5), $W_1$ ...
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0answers
30 views

The orthogonal projection of u onto v is 0?

u = $\begin{bmatrix}-1\\1\\0\\1\\0\end{bmatrix}$ and v = $\begin{bmatrix}1\\0\\1\\1\\1\end{bmatrix}$ Find proj_v^u. The orthogonal projection of u onto v is equal to (u ∙ v/ v∙v)(v) u ...
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1answer
25 views

Finding the basis for W perp when the null space of the span of W is 0

This question requires me to calculate the basis of W perpendicular. W = Span{$\begin{bmatrix}1\\1\\0\\1\\1\end{bmatrix}$,$\begin{bmatrix}-1\\0\\1\\0\\1\end{bmatrix}$} If I set A = ...