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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2answers
15 views

Do these vectors span $\mathbb{R^2}$?

How do I show that $V_1 = (1,1)$ and $V_2 = (-1, 2)$ , $V_3 =(1,4)$ span $\mathbb{R^2}$? I thought you can express them as a linear combination equaling some fixed $x_1$ and $x_2$, but then I get ...
0
votes
1answer
19 views

How to find a basis for the solution space of a linear system?

How to find a basis for the solution space of this linear system? $$ \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ Solutions ...
-1
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3answers
24 views

Linear combinations and spanning solution spaces

PLEASE help on part A only :) V = Column vectors [1 0 -1] and [1 3 0]. a) Make a system of 3 unknowns and 3 equations of which he solution space is spanned by V? b) Express [1 2 3] as a linear ...
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1answer
25 views

Find the flaw - Sum of two subspaces is a subset of their union

Here is a flawed proof for $V+W\subseteq V\cup W,$ where $V$ and $W$ are subspaces of $\mathbb{R}^n$: Consider $\mathbf{x}\in\left(V\cup W\right)^\perp.$ This implies: ...
0
votes
2answers
37 views

Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 ...
0
votes
1answer
29 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
0
votes
2answers
16 views

If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
1
vote
1answer
15 views

Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
0
votes
1answer
28 views

prove linear independence of polynomials

Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent. I don't ...
1
vote
1answer
23 views

Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? Note that ...
0
votes
0answers
21 views

If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
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0answers
9 views

calculating position of a point knowing two reference lengths

Hi, I would like to know if there is a way to calculate a unique position for Point A knowing the lengths l1 and l2 which are variable string lengths. Point A can move within the range shown below. ...
0
votes
3answers
24 views

When talking about a normed vector space, does it's metric always need to be the induced one?

The title basically says it all. If we have a normed vector space, is it possible to work with the space as a metric space with a different metric than the induced one? So if the space is $(X,||\ ...
1
vote
0answers
30 views

Show that $f:V\to W$ is a $(1,1)$-tensor

I'm currently reading Nakahara's Geometry, topology and physics (about tensors), and came across with the following proposition (exercise 2.12, p.99): Show that a linear map $f:V \to W$ is a (1,1) ...
0
votes
3answers
28 views

Finding subspace's base

Let W be a subspace of $\mathbb{R}^4$: $ \begin{cases} x_1+2x_2+3x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ Find base of W and extend it to the base of $\mathbb{R}^4$ How to approach this ...
0
votes
1answer
29 views

Transforming Vectors

Let $T$ be the linear transformation from $\mathbb{R}^3$ to $\mathbb R^3$ that reflects every vector about the $xy$-plane and then triples its length. How do I find the matrix for $T$?
2
votes
1answer
37 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
0
votes
1answer
25 views

Given multiple polynomial equations find a basis.

I have read several other threads on Math.SE, including the similarly titled: basis of the polynomial vector space I've also checked out a video lecture on Youtube by njwildberger, but I simply have ...
0
votes
1answer
44 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
38 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
21 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
25 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
25 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 ...
5
votes
2answers
42 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
28 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.
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votes
0answers
27 views

Do prove in vector space (about span and subspace)

(a) let vector u = (a,b,b,0) and vector v = (0,c,-c,d) because u dot v = 0, thus v = c(0,1,-1,0) +d (0,0,0,1) therefore, W2 = span {(0,1,-1,0),(0,0,0,1)} the basis for w2 is (0,1,-1,0),(0,0,0,1) for ...
0
votes
1answer
18 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
25 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
15 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
26 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
30 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
37 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
19 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
86 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
39 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
27 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
57 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
27 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
17 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
27 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
11 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
34 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
35 views

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

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
29 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
2answers
39 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 ...