Tagged Questions

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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3answers
43 views

$I+A^*A$ is non-singular whenever $A$ is a square matrix with complex entries ? [on hold]

Let $A$ be a square matrix with complex entries , then is it true that $I+A^*A$ is non-singular ?
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2answers
13 views

Determining whether sets of vectors form a basis

Is $\{(1,1,0,0),(0,0,1,1)\}$ a basis for the subspace of $\mathbb{R}^4$ consisting of all vectors of the form $(a,a+b,b,b)$ with $a,b\in \mathbb{R}$? Here is how I proceeded: First note that ...
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3answers
48 views

Linear Algebra: What do vector spaces represent?

I understand what a vector can represent. But I still don't understand what a vector space represents. I understand that you can add two vectors and that becomes a vector space. What else can you do ...
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1answer
24 views

Vector subspace of polynomials

If I have a set of polynomials of degree at most $2$, such that $p(x) \geq 0$ for any real $x$. It isn't a vector subspace because I can multiply by a negative number such that $p(x) < 0$?
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0answers
9 views

Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses?

I was reading Linear Algebra Done Right. I came across the following question (Ch-1, Q12), for which I have solution , but I am having little confusion regarding it: Q12. (a) Does the operation of ...
0
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1answer
27 views

Prove that B is vector subspace [on hold]

In $ℝ^∞$ we've got $B=\{\{x_n\}^{\infty}_{n=0}| \exists k\in~ (0,\infty) \forall i\in \mathbf{N}, | x_i|\leq k^{-1}_i\}$. Prove that B is vector subspace.
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2answers
30 views

Are all vector spaces also a subspace?

I am currently learning about vector spaces and have a slight confusion. So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list ...
1
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2answers
56 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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1answer
25 views

Need help to understand a line of a proof of diagonalizability of real symmetric matrices

I was reading a proof of diagonalizability of real symmetric matrices using the concept of generalized eigenvalues and understood all except the very starting (and fundamental) line of the proof " if ...
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5answers
263 views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
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1answer
13 views

can't get a solution to a vectors question

We have : $|\vec{u}|=1 ,|\vec{v}|=5 ,\vec{u}\cdot\vec{v}=3$ $\vec{w}=(\vec{u}-\vec{v})\times(2\vec{u}+\vec{v})$ we have to calculate $|\vec{w}|$ after playing with equations we have in our ...
1
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2answers
54 views

Let ${v_1,v_2,…,v_n}$ be a linearly independent subset of a vector space V where $n \geq 4.$ [on hold]

Let ${v_1,v_2,...,v_n}$ be a linearly independent subset of a vector space $ V $where $n \geq 4.$ Set $w_{ij}=v_i-v_j.$ Let $ W $ be the span of $\{w_ {ij} : 1\leq i,j \leq n\}$. Then which is correct ...
0
votes
0answers
29 views

Need some help calculating vectors.

What I am attempting to do is use some math that is frankly beyond me to draw a line in three dimensional space. I am using vector subtraction to create an imaginary line from point A to point B ...
0
votes
1answer
21 views

Is this a valid operator norm?

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm? (I think it is. As it satisfies ...
0
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0answers
13 views

Product of dot products of two vectors

I have a product of innerproduct/dot product of two vectors. $ \langle u_i,v_j \rangle\cdot\langle x_i,y_j\rangle$. Is there any transformation/decomposition such that I can combine $u_i$ with $x_i$ ...
0
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1answer
21 views

How to show this iff relationship?

Let $M$ be a linear operator in $L(V)$, where $V$ is a vector space. How to show $\ker(M^k) = \ker(M^{k+1})$ iff $R(M^k) \cap \ker(M) = \{0\}$. It is not difficult to show $\ker(M^k) \subset ...
2
votes
1answer
15 views

Proving that a set of vectors is a basis in P_3

I want to show that the following set constitutes a basis for the vectorspace of polynomials up to degree 3, i.e. $P_3$: {(t-1),(t+1),(t-1)^3,(t+1)^3)} Since $P_3$ is four dimensional, I believe it ...
0
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0answers
12 views

volume parallelepiped-linear algebra

So I have this exercise where they give me the vertices and i must chose those to use to calculate the volume= absolute value of the transformation matrix. The matrix will be $3\times 3$, so i am ...
2
votes
3answers
53 views

What is the mathematical meaning of $xx^T$

Assume x is a n by 1 column vector, then , it is easily known that $x^Tx$ is the sum of squares. When calculate $xx^T$, it will give you a n by n matrix. I am wondering what is the mathematical ...
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0answers
16 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...
1
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3answers
29 views

How did my book see the rank and what is wrong with my null space

Reduced Row Echleon form: $$\begin{bmatrix} 3 & 1 & 3 & -2 \\ 2 & -1 & 4 & -5 \\ 0 & 0 & 0 & \theta+6 \\ 0 & 0 & 0 & \theta +6 \end{bmatrix}$$ ...
2
votes
2answers
38 views

What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$?

What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$? I think that the answer is $2$, because $\mathbb R^2 \cong \mathbb{C}$ (since all complex numbers can be mapped to $\mathbb ...
0
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1answer
54 views

Find the dimension of $W_{A}$

Let $A\in\operatorname{M}_{10}(\mathbb C)$. Let $W_{A}$ be the subspace of $\operatorname{M}_{10}(\mathbb C)$ spanned by $\{ A^n \mid n\geq 10 \}$ . What about the dimension of $W_{A}$?
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1answer
16 views

How to show that the following subset os $\mathbb{R}^3$ is a linear subspace of the real vector space $\mathbb{R}^3$ iff $a=0$? [on hold]

Show that the subset $v_{0}:=\left\{ \left( x_{1},x_{2},x_{3}\right) \in \mathbb{R}^{3}|x_{1}+x_{2}+x_{3}=a\right\} ,\left( a\in \mathbb{R} \right)$ of $\mathbb{R}^3$ is a linear subspace of the ...
2
votes
1answer
46 views

Does all Eigenvectors of $A$ lie on the vector space of $Ax$?

The problem is with the last part of the following question: I will write my results to the first parts which are correct here : Three Eigenvalues: $$\lambda_1=1 , \lambda_2=2 , ...
1
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1answer
38 views

Bases of complex vector spaces and the axiom of choice

In Zermelo-Fraenkel set theory $ZF$ consider the following statement defined for every field $K$: $B_K$ : Every vector space over $K$ has a basis. It is well-known that $AC \Rightarrow \forall K ...
0
votes
1answer
18 views

How does field influence the dimension of vector space?

The dimension of a vector space is the number of its basis. And this field is defined over a field. I am figuring out how a field influence the dimension of vector space. For example ...
2
votes
2answers
44 views

Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.

I'm trying to prove the following statement: In a vector space $V$ over a field $\mathbb{F}$, if $\{u,v,w\}$ is a basis for $V$, then $\{u+v, u+v+w, v+w\}$ is also a basis. $$\underline{\text{My ...
0
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0answers
25 views

Determining the Dimension of a Vector Space [on hold]

I have to determine the dimension of the vector space, and as given in the pic, I need to use the concept of equivalent norms.
0
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2answers
15 views

Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism.

Let V, W be finite dimensional vector spaces. Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism. Thanks in advance! I ...
0
votes
1answer
16 views

$T$ on $\mathbb R^n$ is diagonalizable over $\mathbb R$ with $T^k = I$ , for some $k$ , then is $T^2=I$ [on hold]

If $T$ is a diagonalizable (over $\mathbb R$ ) linear transformation on a finite dimensional real vector space such that $T^k$ is the identity operator for some positive integer $k$ , then is it true ...
1
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1answer
19 views

$T$ is diagonalizable on finite dimensional v.s. $\implies$ $(T^2+T+I)(\vec v) \ne \vec 0 , \forall \vec v \ne \vec0$?

Let $T$ be a diagonalizable (over $\mathbb R$) operator on a finite dimensional real vector space ; then is it true that there is no non-zero vector $\vec v$ such that $(T^2+T+I)(\vec v)=\vec 0$ ? ...
2
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4answers
30 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
0
votes
1answer
13 views

Question about the formula of lines and planes

I just watched some videos from Khan Academy which left me slightly confused. I don't really understand why the formula of a plane is $n(p_1-p_2) = 0$, where $n$, $p_1$ and $p_2$ are vectors. I do ...
10
votes
2answers
269 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
4
votes
1answer
39 views

Is this sufficient for continuity?

Assume you have a map $\phi : V \rightarrow \mathbb{C}$, where $V$ is a complex vector space. Now, if we have $\phi(\lambda x) = |\lambda | \phi(x)$ and $\phi(x+y)^2+ \phi(x-y)^2 = 2\phi(x)^2 +2 ...
3
votes
1answer
25 views

Tensor product of linear transformations

If $U$ and $V$ are finite-dimensional vector spaces then $U^*\otimes V^* \approx (U \otimes V)^*$ via the isomorphism $\tau: U^*\otimes V^* \to(U \otimes V)^*$ given by $\tau(f \otimes g)(u \otimes v) ...
0
votes
0answers
17 views

Jacobian equals the product of scale factors

I have to prove that in 2 dimensions $J(\frac{x,y}{q_1,q_2})=h_1 h_2$ (1), where $q_1, q_2$ are the new mutually perpendicular coordinates and $h_1, h_2$ are the respective scale factors (exercise ...
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0answers
15 views

Changing Bases of Linear Transformations

Let $\mathcal{U}$ be a vector space with basis $\mathcal{B}_1=\{u_1,\cdots,u_k\}$ and let $\mathcal{V}$ be a vector space with basis $\mathcal{B}_2=\{v_1,\cdots,v_k\}$ and let $\mathcal{W}$ be a ...
1
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0answers
21 views

orthogonal complement of the linear span

I am going back through old problems, and I still can't figure this one out. Find an orthonormal basis for the orthogonal complement of the linear span of the vectors $(1,0,1,0)$ and $(1,0,-1,0)$ in ...
0
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1answer
56 views

Help me perfect out my current linear algebra knowledge

My questions are along my workings, I have attempted both the parts as much as possible as I can. Please help me on this question. My question comes as (i) Is my proof perfect? (ii) Am I correct? ...
0
votes
1answer
14 views

How come that sum of two lines in a triangle equals the third line?

I have been looking into this question and cannot understand how they came to a conclusion in their solution. we know that : $$(1) :\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 $$ ...
0
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0answers
28 views

For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?

Is it true that for $n \ge 2$ , every linear operator $T$ on $\mathbb R^n$ has an invariant subspace of dimension $2$ ? I know that $T$ always either have a $1$ or $2$ dimensional invariant subspace ; ...
1
vote
1answer
15 views

A question on nilpotent linear operator on finite dimensional vector space with dimension same as degree of nilpotency

Let $T$ be a nilpotent linear operator of index $n>1$ ($T^n$ is the null operator but $T^{n-1}$ is not ) on a vector space of dimension $n$ ; then how do we prove that there is no linear operator ...
0
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0answers
27 views

Basis of a set of vectors

Assume that we have a set of vectors as $S=\{v|v=(\sum_{i=r+1}^nx_ic^i_1,\dots,\sum_{i=r+1}^nx_ic^i_r,x_{r+1},\dots,x_n)^T,\forall x \in \mathbb{R}^{n-r}\}$. Here the first $r$ elements of the vector ...
1
vote
4answers
33 views

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$.

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$. I'm not sure how to start proving this. I don't ...
0
votes
1answer
22 views

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation.

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation. I'm sorry, but this question throws me off in many ...
0
votes
1answer
27 views

Linear map - how to show this?

Assuming that I have a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ and we have $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$. Is this sufficient to conclude that $A( \lambda x+y ,x ) = \lambda ...
1
vote
1answer
18 views

Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
1
vote
2answers
32 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...