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 ...

learn more… | top users | synonyms

0
votes
1answer
23 views

$\mathbb R^2$ as a plane

What elements allow me to say that $\mathbb R^2$ can be seen simply as a plane (or not if that is the case)? Yes, $\mathbb R^2$ is a vector space (not only with that characteristic) with multiple ...
1
vote
2answers
31 views

Proof that every subspace is a vector space

I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that: $$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$ ...
0
votes
1answer
14 views

Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
0
votes
1answer
20 views

Prove that the vectors u,v,w are orthonormal in V

Let V be the vector space R3 with inner product (v,w)=3(v1w1)-2(v1w2)-2(v2w1)+5(v2w2)-3(v2w3)-3(v3w2)+3(v3w3) where v=v1,v2,v3 and w=w1,w2,w3 Prove that the vectors u=(1,1,1), ...
1
vote
1answer
21 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
4
votes
2answers
36 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
-1
votes
1answer
27 views

Proof of Vector Space Axioms [on hold]

Where can I find detailed proof of vector space axioms? Any reference to a book, website or video lecture.
-5
votes
0answers
19 views

Do prove in vector space (about span and subspace) [on hold]

How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part ...
0
votes
1answer
14 views

A simple question related to One-to-One function and linear operator

I was stuck in one line derivation about the linear operator-related question: Suppose $T$ is linear operator maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. and let $c>0$ be constant. If for all ...
2
votes
1answer
12 views

Possible values of nullity in 4x2 matrix

Let $A$ be a 4 by 2 matrix. Explain why the rows of $A$ must be linearly dependent. What are the possible values of nullity(A)? I understand the first part. I do not understand the second part. The ...
0
votes
1answer
17 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
0
votes
0answers
13 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
0
votes
0answers
22 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
0
votes
2answers
18 views

arc length, problems to find the limits for t

How do I find the limits for t? (a) Let $C$ be the parametric curve $$r(t) = \frac{t^3}{3}\hat i + t^2\hat j + 2t \hat k$$ Determine the arc length of $C$ between the points $(0, 0, 0)$ and $( 1/3, ...
0
votes
2answers
28 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
2
votes
2answers
31 views

Need some help on linear algebra Subspace test

Any help would be appreciated, thank you.
0
votes
2answers
17 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
0
votes
1answer
18 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
0
votes
3answers
79 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
0
votes
1answer
36 views

$T$-invariant subspace and minimal polynomial

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
0
votes
3answers
25 views

Canonical isomorphism between $V$ vector space and its second dual $V^{\circ \circ}$

I came a across this when I was reading some book. It says let $V$ a finite dimensional vector space of some field and there is a canonical isomorphism $\phi$ between $V$ and $V^{\circ \circ}$ but ...
-1
votes
2answers
51 views

Find the projection p of x onto the span of u1 and u2

where $u_1=(2/3, 2/3, 1/3)$ and $u_2=(1/\sqrt2, -1/\sqrt2, 0)$ and $x=(1,2,2)$ how do I find the span of $u_1$ and $u_2$? after that do I just use the formula for the vector projection of x onto the ...
0
votes
2answers
25 views

Write the Jordan form of an operator

These are the properties that apply to the operator $A$. $k_A(x)=x^4(x-2)^4, d(A)=2, d(A^2)=4, d((A-2I))=2, (d((A-2I)^2)=3$ $d$ denotes the defect. $k_A$ is the characteristic polynomial. I ...
0
votes
1answer
14 views

Finding base of a subspace

Find base of a subspace and expand it to the base of $\mathbb{R}^4$ subspace is given by the following system of eqiuations: $ \begin{cases} x_1+2x_2+2x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ ...
0
votes
1answer
24 views

Disprove that this subset of P3 is not a subspace by using a counterexample

The set of all polynomials with degree 3 plus the zero polynomial. A hint would be appreciated to get me going :)
0
votes
0answers
10 views

Show subspace can be rewritten as $n-k$ equations

Prove that every $k$ dimensional subspace $V \subset K^n$ can be described using $n-k$ linear equation. I think about applying Kronecker-Capelli theorem.
2
votes
1answer
33 views

Finding the Jordan basis of a linear map

A linear map $A$ is given in the canonical basis with the matrix $$ \begin{bmatrix} -2&0&-2&-2\\ 1&0&1&1\\ -1&1&-1&-1\\ 3&-1&3&3\\ \end{bmatrix} $$ ...
-1
votes
0answers
33 views

The set of matrices with nonnegative determinant is not a subspace. [on hold]

Disprove using a counterexample: The set of all $3\times 3$ matrices with determinant $\ge 0$ is a subspace of $M_3(\Bbb C)$.
0
votes
1answer
19 views

Let T:V->W be linear, show KerT is a subspace of V and imT=T(V) is a subspace of W

Ok so I have already proven that KerT is a subspace of V, which is pretty obvious because the kernel is just the 0's, though I'm not sure I did it formally enough. The second part I don't know how to ...
1
vote
1answer
27 views

show there exist non zero vector which is linear combination of other

sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non ...
0
votes
3answers
37 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
0
votes
0answers
15 views

Proof Explanation: Vector Space of Polynomials with Average Value 0 around a circle

The question is from Putnam 2009 B4. Problem: Say that a polynomial with real coefficients in two variable, $x,y$, is balanced if the average value of the polynomial on each circle centered at the ...
0
votes
3answers
24 views

Determining if a set is in the subspace of a continuous function

Let $A={\rm span}\{\cos^2x,\sin^2x\}$ be a subspace of the set of functions $C[0,\pi]$, for each of the following functions in $C[0,\pi]$, determine whether or not it is in $A$. $f(x)=1$ ...
1
vote
2answers
63 views

Prove $W \cap W^\perp =\{\vec{0}\}$

If $W$ is a subspace of $\mathbb{R}^n$, then $W^\perp = \overline{W} = \{v \cdot w = 0, \forall w \in W\}$ Prove $W \cap W^\perp = \{\vec{0}\}$. How do I fully prove this intersection is ...
0
votes
1answer
23 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
1
vote
1answer
14 views

Find all unit vectors in the plane determined by vectors u and v that are perpendicular to the vector w.

Find all unit vectors in the plane determined by vectors u=(0,1,1) and v=(2,-1,3) that are perpendicular to the vector w=(5,7,-4). This is the question. I found the plane that determined by u and v, ...
1
vote
0answers
34 views

Check if set of functions is a basis of space

Let $f_a \in R^R$ be function given by $f_a(x)=1$ if $x=a$ and $f_a(x)=0 $ if $x \neq a$ for $a \in R$ Decide if set of functions $f_a$ is a basis of space of functions $R^R$ ? I think I know how to ...
2
votes
1answer
18 views

Do two isomorphic finite field extensions have the same dimension?

If $E = F(u_1, \cdots u_n) \cong \bar{E} = F(v_1, \cdots v_m)$ then do the two extensions necessarily have the same dimension over $F$?
3
votes
0answers
25 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
-4
votes
1answer
45 views

Verifying the axioms of a vector space for $V =\{(a,b):a,b\in\mathbb R \}$ with unusual scalar multiplication [on hold]

Let $V =\{(a,b):a,b\in\mathbb R \}$. Addition in $V$ is $(a_1,b_1) +(a_2,b_2) = (a_1+a_2, b_1+b_2)$ and scalar multiplication is $k(a,b) = (ka, 0)$. Is $V$ a vector space? Why? I'm mostly lost ...
0
votes
0answers
17 views

1st isomorphism theorem for linear transformations (algebra)

For a field K, U' and U'' are vector subspaces of a vector space U over K. It needs to be proven that the transformation φ: U' →(U' +U'')/U'', u' 􏰀→u' +U'', is a surjective linear transformation, ...
1
vote
0answers
51 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with ...
2
votes
1answer
33 views

Which Field Operators Construct the Vector Space

Question 14 in F-I-S section 1.2 asks: Let $\mathbf{V}=\{(a_1,a_2,\ldots ,a_n)\colon a_i\in \mathbb{C}$ for $i=1,2,\ldots n\}$; so $\mathbf{V}$ is a vector space over $\mathbb{C}$. Is $\mathbf{V}$ ...
0
votes
1answer
16 views

“Absolutely equal” linear functionals and collinearity

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb C$ and let $X^*$ denote its dual (i.e., the space of all continuous linear complex-valued functions over $X$). Suppose that $f,g\in X^*$ ...
1
vote
1answer
70 views

Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$. I know that having the ...
0
votes
0answers
32 views

Independence in Banach space

Everyone knows one of the basic theorems in linear algebra: $k+1$ vectors can't be linear independent in the span of $k$ vectors. Also, it's pretty easy to prove that there is no uncountable system of ...
0
votes
0answers
7 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
0
votes
0answers
9 views

Why is the following progression allowed?

I'm looking at a proof of a thing related to vector spaces and in that proof we have the following progression: x$^{H}$($\alpha$y + $\beta$z) = $\alpha$x$^{H}$y + $\beta$x$^{H}$z where x, y, z are ...
0
votes
5answers
51 views

linear independence with $\sin x, \cos x$

I don't know why $\sin x$ and $\cos x$ are lineary independent since if we take linear combination $a\cdot \sin x + b \cdot \cos x=0$ and for $a=\sqrt{3}$ and $b=1$ and $\displaystyle x=\frac{\pi}{6}$ ...
0
votes
1answer
16 views

Finding vector when conditions are given

Given subspace (of $\mathbb{R}^4$) $V= \rm span ([2,3,1,2], [3,2,2,3], [1,-1,1,1]) $ For $\beta_1=[1,1,1,1], \beta _2=[2,-1,1,2]$ desribe set of all vectors $[b_1, b_2] \in \mathbb{R}^2 $ such that ...