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
173 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
36 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
53 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
56 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 :)
2
votes
1answer
50 views

Finding the Jordan basis of a linear map

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

Let $T:V\to W$ be linear, show $\ker T$ is a subspace of $V$ and $\operatorname{im} T = T(V)$ is a subspace of $W$

OK, so I have already proven that $\ker T$ 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 ...
1
vote
1answer
41 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 ...
1
vote
2answers
48 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
3answers
48 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 $A$. $f(x)=1$ ...
2
votes
2answers
206 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
40 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
267 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
104 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
42 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
1answer
89 views

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

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 ...
2
votes
1answer
47 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
28 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
240 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
41 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
5answers
88 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
17 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 ...
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votes
2answers
91 views

Is the statement “the empty set is a subspace of every vector space” true of false?

Is this statement true or false, and why? The empty set is a subspace of every vector space.
1
vote
1answer
36 views

Find $t$ such that (subspace)

For $t\in\mathbb{R}$ a subspace of $\mathbb{R}^4$ is given $V_t={\rm span}([t, -1, 2,-1],[1,-1,-1,1])$. Find all $t\in\mathbb{R}$ such that $V_t={\rm span}([4, -3, 0,1],[1,0, 3,-2])$. EDIT: Still ...
-1
votes
1answer
22 views

Finding base vectors

How to find base vectors of such a subspace given by the following equation: $W=\{[x_1,x_2,x_3,x_4] \in\mathbb{R}^4 : 2x_1+x_2-x_3-x_4=0 \}$
1
vote
0answers
53 views

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$. I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$. I also ...
0
votes
1answer
22 views

Let $S$ be a vector space. I want to show that for every subspace $U$ of $S$, the closure $\bar U$ is again a subspace of $S$.

Suppose $S$ is a vector space with norm $|| \ ||$ and $\rho$ the corresponding metric. I want to show that for every subspace $U$ of $S$, the closure $\bar U$ is again a subspace of $S$. I've proven ...
1
vote
2answers
33 views

What does dimension of polynomial mean?

So, I know that the vector space of polynomials with degree $n$ has dimension $n+1$. What does this exactly mean? I'm asking specifically because of the following question (from Putnam and Beyond): ...
0
votes
1answer
48 views

To prove $V$ Is not a vector space and my attempt

Let V be set of all pairs (x,y) of real numbers , And F b field of real numbers Define $c(x,y)$ = $( |c|x , |c|y)$ Addition defined as usual but it is not main concern here What i have done is ...
0
votes
1answer
44 views

How is this expression well-defined?

I am going through the book "Introduction to Tensor Product of Banach Spaces" by Raymond Ryan. The tensor product of vector spaces is introduced in the first chapter which I briefly outline now. Let ...
1
vote
1answer
19 views

Finding vector from a given subspace

Given the subspace described by those three vectors: $W=lin([1, 0, 2, 4], [0, 1, 2, 3], [0, 1, 0, 1] ) $ Now i want to pick up any vector that is located in this subspace. How should i do this?
0
votes
1answer
40 views

Coordinates of vectors in bases

Two vectors from the standard basis are a = (1,0,1) and b = (1,1,1). What are the coordinates of these vectors in the basis {(1,2,3),(2,3,1),(3,0,1)}. I am not even sure how to answer this question. ...
1
vote
0answers
103 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
1
vote
1answer
73 views

Eigenvalues of linear operator TS and ST for infinite dimensional space

Here is the original problem: Let $S$ and $T$ be linear operators on a finite-dimensional vector space $V$. Show that $TS$ and $ST$ have the same eigenvalues. I can prove it. However, my question is: ...
1
vote
1answer
56 views

show that if S(finite) spans V(finite), (but not basis) then S can be reduced to a basis for V by removing appropriate vectors from S

problem 18: Suppose that S be a nonempty finite set of vectors in a finite dimensional Space V. Use problem 16 to show that if S spans V, but is not a basis for V, then S can be reduced to a basis for ...
0
votes
1answer
45 views

Basis for solution space?

For the matrix: $$ \begin{bmatrix} 1 & 0 & 2 & | & 0 \\ 0 & 1 & 3 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} $$ ...
1
vote
2answers
71 views

Exponential objects in $k$-$\mathbf{FDVect}$

In my differential geometry class we've now moved onto algebraic/differential forms and to begin the section we're doing a quick and easy review of dual vector spaces. On a problem sheet I am ...
1
vote
3answers
97 views

Cardinality of the basis for $\mathbb{R}$ over $\mathbb{Q}$? [duplicate]

This question came up as a discussion I had with a friend. Clearly, the basis is not of finite cardinality since that would imply the set $\mathbb{R}$ has the cardinality $\aleph_{0}$ which is false. ...
1
vote
2answers
82 views

K[x]-module is k-vector space

Could anyone tell me why an $A$-module is a $k$-vector space with a linear transformation? (Here $A=k[x]$ where $k$ is a field.) Thanks!
0
votes
3answers
53 views

How to check that this is an orthogonal linear map with $\det (A) = 1$, so it is a rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
0
votes
1answer
14 views

Calculate 1-norm of a vector using another matrix or vector

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
0
votes
0answers
38 views

How to find the axis of the rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
1
vote
3answers
50 views

Similarity of linear transformations

Suppose there are two linear transformations $A$ and $B$ on the same finite dimensional vector space $V$, such that $\dim Im(A) = \dim Im(B)$. Is it always true that they are similar. What about the ...
2
votes
1answer
53 views

Vector space and algebraic closure of a field

I hope you can help me with these questions, I can't really come up with a solution! Let $V_k$ be a vector space of dimension $n$ over a field $k$. Let $K=\bar k$ be the algebraic closure of $k$. A ...
2
votes
1answer
61 views

The norm of operator $\mathscr{L}$ on the finite-dimensional vector space $V$ equals the norm of operator restricted by some Invariant subspace.

The norm of linear transformation $\mathscr{L}$ on the finite-dimensional vector space $V$ over $\mathbb{R}$ with standard inner product equals the norm of linear transformation $\mathscr{L}$ ...
2
votes
2answers
275 views

Modules of Finite Length over Local Artinian Rings

Let $R$ be a commutative local artinian ring with identity. Denote its maximal ideal by $\mathfrak{m}$ and let $\mathbb{k}$ denote the residue field $\mathbb{k}=R/\mathfrak{m}$. Assume also that there ...
1
vote
2answers
48 views

How to show a particular set $S$ is a basis for $M_{2\times 2}$?

I'm given a set $S$ of four $2\times 2$ matrices with numbers in them and need to show they are a basis for $M_{2\times 2}$. all I know is that the linear combination of these matrices $= [0,0,0,0]$ ...
1
vote
2answers
1k views

Find the equation of the plane spanned by $v_1$ and $v_2$, find a vector $v_3$ that can be added to produce a basis for $\mathbb{R}^3$

Let $v_1=(-1,2,3)$ and $v_2=(5,3,-1)$. Find the equation of the plane spanned by $v_1$ and $v_2$. Also find a vector $v_3$ that can be added to the set $\{v_1,v_2\}$ to produce a basis for ...
0
votes
1answer
36 views

Linear Algebra Spanning question

If I have two 3x1 column vectors in a vector space V that are linearly independent, how can I make a system of 3 eqns whose solution will span V? For example, column vector [1,3,0], column vector ...
1
vote
2answers
36 views

How do I complete a set of three vectors in $\mathbb{R}^4$ to a basis of that space?

Given vectors $$w_1 (0,-1,2,1),\quad w_2 (1,0,2,1),\quad w_4 (2,1,1,0),$$ how do I find another vector $v$ such that $\{w_1, w_2, w_4, v\}$ is linearly independent? My approach is to write them all ...
0
votes
2answers
33 views

finding dimension of subspace of $P_3$ given by $H=\{a+bx^3:a,b \in \mathbb R\}$

I'm not sure how to find the dimension of this set or any set like this based on what I know about dimensions. $H=\{a+bx^3:a,b \in \mathbb R\}$ All it seems I'm given in my notes is that $\dim(P_3) ...