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
26 views

Linear dependence when number of vectors is greater than/less than the dimensions of the vector space

Simple question here, I just need some clarification of a theorem. Theorem: if k > n, then any k vectors in $R^n$ are linearly dependent. Nice and easy I guess! My question is this: Does this imply ...
2
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1answer
44 views

What is the probability of choosing r independent vectors in $\mathbb{R}^n$ in the unit sphere?

I was trying to compute the probability of choosing $r \leq n$ indepedent vectors $a_i \in \mathbb{R}^n$ such that they are independent. I was told that the probability that they are not independent ...
0
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1answer
24 views

Is this a vector space? If not, how can I make it one?

S (2x2 matrix) = {(a, b), (c,1) | $a,b,c$ is in $ \Bbb R$} I know for a vector space we must: 1. Define Addition 2. Define Scalar Multiplication 3. Have a set of numbers 4. Have a Field I know that ...
4
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2answers
130 views

Kreyszig's Functional Analysis Section 2.8: How is the canonical embedding map injective?

Let $X$ be a vector space over the field $K$ of the real or complex numbers. Let $X^*$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all ...
0
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1answer
17 views

Find a finite set of vectors which spans $W$.

Let $W$ be the set of all $(x_1, x_2, x_3, x_4, x_5)$ in $\Bbb R^5$ which satisfy $2x_1-x_2+{4 \over 3}x_3 - x_4\qquad = 0$, $x_1\qquad+{2 \over 3}x_3\qquad- x_5 = 0$, $9x_1-3x_2+6x_3-3x_4-3x_5 = 0$. ...
2
votes
1answer
22 views

Prove $V_{e} + V_{o} = V$

Prove $V_{e} + V_{o} = V$ where $V_{e}$ is a subset of even functions from $R$ into $R$, $V_{o}$ is a subset of odd functions from $R$ into $R$. I have proved $V_{e}$, $V_{o}$ are subspaces and ...
2
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0answers
62 views

What is the $\dim L(X,Y)$?

Let $X$ and $Y$ be two finite-dimensional vector spaces over the same field $K$, and let $L(X,Y)$ denote the vector space of all linear operators $T \colon X \to Y$. Then what is $\dim L(X,Y)$? My ...
1
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1answer
17 views

Show that the vectors form a basis for $R^3$.

Show that the vectors $\alpha_1 = (1, 0, 1)$, $\alpha_2 = (1, 2, 1)$, $\alpha_3 = (0, -3, 2)$ form a basis for $R^3$. Is it enough to show that the vectors are linearly independent?
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3answers
24 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...
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0answers
21 views

Proving properties about cofactors when matrix is not invertible

(1) $cof(A^t) = cof(A)^t$ (2) $cof(A)^t = det(A)I$ I have (at least, I think so) proofs of (1) and (2). But the proofs require the matrix $A$ to be non-singular. How do I prove (1) and (2) if $A$ is ...
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3answers
6k views

Show that V is a subspace of M2x2 Matrices and Determine a basis

A bit of information to start us off: Let V denote the set of all 2x2 matrices with equal column sums. Show $V$ is a subspace of $M_{2\times 2}$ matrices: and.... ...
0
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1answer
46 views

Gram-Schmidt procedure on functions

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials. Here i what i understand If i have 2 functions, say ...
0
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0answers
19 views

Problems with the hypothesis in a fixed point theorem

Leray-Schauder fixed point theorem : If $D$ is a non-empty , convex , bounded and closed subset of Banach space $B$ and $T:D \to D$ a compact map , then $T$ has a fixed point in $D$. I have ...
1
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3answers
70 views

If M,N are finite dimensional vector spaces with same dimension ,then if M is subset of N ,then M=N

If M,N are finite dimensiona;l vector spaces with same dimension then if M is subset of N ,then M=N I think i need to show that both vector spaces are spanned by same bases in order to do this or to ...
0
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1answer
43 views

$\mathfrak{sp}_4$ is a subspace of the vector space of all $4\times 4$ matrices

Let $\mathfrak{sp}_4$ denote the set of all matrices $X$ satisfying $$X^TM+MX=0$$ How can I show that $\mathfrak{sp}_4$ is a vector subspace of the vector space of all $4\times 4$ matrices? I ...
3
votes
3answers
114 views

Prove that odd polynomials $f(x)$ of degree $\leq 10$ with $f(-1) = 0$ form a vector space.

Let $P(X)$ be the usual vector space of polynomials in $x$ with real coefficients. Let $U$ denote the subset of $P(X)$ consisting of those elements $f(x)$ which have degree less than or equal ...
7
votes
4answers
571 views

Why is a function space considered to be a “vector” space when its elements are not vectors?

I am confused by the notion of a function space. For example consider the basis $\{1, x, x^2\}$ which is the basis for the vector space of all polynomials of degree at most $2$. What is the notion of ...
0
votes
1answer
26 views

Prove column space is a subspace of $\mathbb{R}^n$

I have an exercise on my last assignment for linear algebra, which is the following: The column space $C(A)$ of linear mapping $A: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is defined by: ...
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0answers
12 views

count of non-overlapping vectors for finite vector space

Consider 1: How many unique vectors can be drawn from the origin to a nxn grid of dots? Or v=(x,y) where 0<= x,y < n and x,y are natural numbers. Answer: Count = n² - 1 (not including nul ...
1
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1answer
102 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
2
votes
2answers
55 views

Why the column space of a matrix is useful?

I know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture: ...
2
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1answer
17 views

Prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$

I am trying to prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$ For U is nonempty I have: Let $u(x) = 0x^4 + 0x^3 + 0x^2 + 0x + 0$ For U is closed under $+$ I have: Let $x, y ...
4
votes
2answers
42 views

How to find if this huge vector is in the column space of this huge matrix?

I newbie to linear algebra, so I hope you are patient with me. I have to say if a vector $\vec{u} = \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1\end{matrix} \right]$ is in the column space ...
2
votes
2answers
47 views

To prove any two basis of Finite Dimensional Vector Space have same number of elements

To prove any two basis of Finite Dimensional Vector Space have same number of elements If i take bases as $S_!$ = {$\alpha_!$ ,$\alpha_2$ ,....$\alpha_n$ } $S_2$ = {$\beta_!$,$\beta_2$ .... ...
0
votes
2answers
21 views

existence of Matrix as an isomorphism

let $U$ be a subspace in $F^n$, let $V$ be a subspace in $F^m$, prove the existence of an $m\times n$ matrix $A$ such that $\text{Row}(A) = U$ and $\text{Col}(A) = V$ ok, um I forgot to mention ...
1
vote
1answer
19 views

Showing there exists an $m\times n$ matrix $A$ with $\text{Row}(A) = U$ and $\text{Col}(A) = V$ where $U$ and $V$ are subspaces.

Let $F$ be a field. Let $U$ be a subspace of $F^n$ and let $V$ be a subspace of $F^m$. Suppose that $\dim U = \dim V$. Then there exists an $m\times n$ matrix $A$ with entries in $F$ such that ...
1
vote
1answer
38 views

How is this map injective?

Let $X$ be a (real or complex) vector space, let $X^{*}$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all linear functionals defined on ...
0
votes
3answers
53 views

Commutativity of scalar/vector product: $a\mathbf{v}=\mathbf{v}a$ for all $a \in F$ and $\mathbf{v} \in V$

There are traditionally 8 axioms to check whether a set $V$ together with a field $F$ constitute a vector space. A common list of axioms can be found here. Missing from the list, however, is a ...
1
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0answers
51 views

Dimensions of spaces over different fields

We view $\Bbb C_2 = \{{w \choose z}:w,z\in \Bbb C\}$ as a vector space over $\Bbb C$, $\Bbb R$ and $\Bbb Q$. Let $x_1={i \choose 0}$, $x_2={\sqrt2 \choose \sqrt5}$, $x_3={0 \choose 1}$, $x_4={i\sqrt3 ...
0
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3answers
38 views

The only dense linear subspace of $\mathbb{C}^n$.

I want to prove this question: the only dense linear subspace of $\mathbb{C}^n$ is $\mathbb{C}^n$ itself. My immediat attempt is think of $\mathbb{C}^n$ as a closed subspace of itself, so it is its ...
0
votes
0answers
18 views

What is the solvability condition for the equation Px=b?

Suppose $P$ projects vectors in $\mathbb{R}^n$ orthogonally into a linear manifold $M$. What is the solvability condition for the equation Px=b? the hint that is given: b must be in the range of P ...
0
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2answers
322 views

proof of the full exchange lemma

Let V be spanned by $\{v_1,...,v_k\}$ and let $\{u_1,...,u_k\}$ be a linearly independent subset of V, then: 1) $k\leq n$ 2) $\exists$ a spanning set $\{w_1,...,w_n\}$ for V where $w_i = u_i$ for $1 ...
4
votes
1answer
33 views

Basis for proper rational functions

Suppose $F$ is a field, and let $F(x)$ denote the $F$-vector space of all rational functions $\frac{f(x)}{g(x)}$, where $f,g\in F[x]$ are polynomials, with $g$ different from zero. Let $F(x)_p$ denote ...
1
vote
1answer
20 views

Find an orthogonal matrix such that its first line is $\frac{1}{5},\frac{2}{5}$

An orthogonal matrix is one matrix $A$ such that $A^t = A^{-1}$. So what I did: Suppose: $$A = \begin{bmatrix}\frac{1}{5}&\frac{2}{5}\\x&y\end{bmatrix}$$ Then: ...
0
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0answers
10 views

Find an isometry such that its change of basis matrix is… [duplicate]

I need to find the isometry such that the change of basis matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\0 & 0 & 1\\x & ...
0
votes
1answer
27 views

Is it defined the product of vectors of different spaces?

I know that the sum of vectors of different spaces is not defined, but what about the multiplication of vectors of different spaces. For example, what about the multiplication of $v_1 = ...
1
vote
1answer
28 views

Find a isometry such that the matrix in respect to the canonical basis is:

I need to find a isometry such that the matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\0 & 0 & 1\\x & y & ...
17
votes
4answers
414 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
0
votes
1answer
22 views

Complete the following proof that $-u$ is the unique vector in V such that $u+(-u)=0$.

suppose that $w$ satisfies $u+w=0$. Adding $-u$ to both sides we have $(-u)+[u+w]=(-u)+0$ $[(-u)+u]+w=(-u)+0$ $0+w=(-u)+0$ $w=-u$
1
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2answers
33 views

Orthogonal complement of subspace $W = span(5,1+t)$

I have this subspace of $P_2(\mathbb R)$ and I need to find its orthogonal complemente, using the inner product defined as $$<p(t),q(t)> = \int_o^1 p(t)q(t) dt$$ So I'm assuming the vector $$v ...
4
votes
2answers
101 views

Good article (or book) about coordinate-dependent linear algebra, for those already familiar with coordinate-free aspects.

I have a decent understanding of coordinate-free linear algebra. For example: (not-necessarily-finite-dimensional) vector spaces, linear transforms, (possibly infinite) products of vector spaces, ...
1
vote
1answer
33 views

Orthonormalization of basis ${1,1+t,2t^2}$ by $\langle u,v\rangle = \int_{0}^{1}uvdt$

Im doing the process: $$g_1 = \frac{1}{||1||} = 1\\v_2 = 1+t-<1+t,1>1 = 1+t-\int_{0}^{1}(1+t)1dt = t-\frac{1}{2}\\g_2 = \frac{v_2}{||v_2||}$$ but $$||v_2|| = \int_{0}^{1}t-\frac{1}{2}dt = 0$$ ...
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votes
2answers
43 views

linearly dependent or linearly independent [closed]

Is it true that if $\vec{u_1}$ , $\vec{u_2}$ and $\vec{u_3}$ are linearly dependent then $\vec{v_1}$ = $\vec{u_2}$ + $\vec{u_3}$ , $\vec{v_2}$ = $\vec{u_1}$ + $\vec{u_3}$ , $\vec{v_3}$ = $\vec{u_1}$ + ...
1
vote
1answer
49 views

Closest Vector in a Inner Product Space

Let $V$ = $\mathbb{R}^n$ Note that $\langle -,-\rangle$ defines the Inner Product on $\mathbb{R}^n$ $$\|v\| = \sqrt{\langle v,v \rangle}$$ Consider the standard Distance Function $$d(x,y) = ...
1
vote
0answers
20 views

Do affine spaces have coordinate transformations?

I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was ...
1
vote
1answer
31 views

Clarification between a module and a vector space?

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.5$ it talks about Module and Vector Spaces: (1) If $K$ is a commutative ring with identity, a module over $K$ ( or a $K$-module) is ...
1
vote
2answers
33 views

What would be a characterization of a definite operator?

Let $V$ be an $n$-dimensional inner product space and let's call $T\in \mathcal L (V)$ definite if $$\forall x \neq0: \langle Tx,x\rangle \neq 0. $$ An obvious sufficient condition for $T$ to be ...
3
votes
1answer
33 views

Equation of a plane containing a straight line

A straight line passes through the points $(1, 2, 3)$ and $(-3, -2, -1)$. I have calculated the system of equations of this line to be $$ x = 1 - t,\, y = 2 - t, \, z = 3 - t $$ The question I ...
1
vote
1answer
41 views

Vector Space Subspace Proof

Suppose that W is a subspace of a finite-dimensional vector space V . Prove that W = V if and only if dim W = dim V. This is what I did: Suppose dim W =dim V$\iff|$basis of $W|=|$basis of $V|$ and ...
6
votes
1answer
282 views

Calculating the intersection of two spaces of polynomials

This problem is driving me nuts. I feel like there should be an elementary argument, yet I have failed to find one. Consider the vector space $V_n=\mathbb Q[x]/{x^{2n+1}}=\mathbb Q\{1,x,x^2,\ldots, ...