# 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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### Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
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### $V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2)$?

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2)$ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement ...
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### Logical dependence between vector space axioms

My question is not very long, but I'd like to explain where it comes from. Consider the classical definition of vector spaces: $E$ is said to be a vector space over a field $F$ when: A) E is a ...
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### log norm inequality for lower triangular part of matrix

Suppose $L$ is the lower triangular part of a matrix $A \in \mathbb{C}^{n\times n}$. Prove that $||L||_2 \leq ||A||_2 \log_2(2n)$. Here $||\cdot||_2$ is the matrix norm induced by the $p=2$ vector ...
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### Question about basis and finite dimensional vector space

I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5) I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
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### What is the relationship between $(u\times v)\times w$ and $u\times(v\times w)$?

Given three vectors $u$, $v$, and $w$, $(u\times v)\times w\neq u\times(v\times w)$. This has been a stated fact in my recent class. But what is the ultimate relationship between them? I would presume ...
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### What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
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### What makes a vector an object with both magnitude and direction?

According to my understanding, A vector is an element of a set called the vector space which satisfies a list of axioms like : closure under vector addition, closure under scalar multiplication, ...
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### Linear algebra - Dimension theorem.

Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$. Dimension theorem states: $$\dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$ My question is: Why is $U \cap W$ necessary in this ...
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### Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?

Can we define a vector space structure on $\mathbb {R}^n$ other than usual scalar multiplication and usual addition such that the dimension of $\mathbb {R}^n$ over $\mathbb {R}$ is not $n$ but some $m$...
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### Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)

I think I solved it, but I used a dirty trick, I'd like someone to review it, that would be great. Let $X,Y$ be linear spaces over field $F$. and $T:X \to Y$ a linear transformation.For each $T$ we ...
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### How to find an intersection of a 2 vector subspace?

Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$. how to get thier intersection?
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### What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
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### Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
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I would really love any sort of proof of this. I have a very elementary geometric proof for $R^{2}$. That's mainly because I can easily represent $cos\theta$ in the form of $\frac{u_{x}v_{x}+u_{y}v_{... 2answers 1k views ### Consistency of matrix norm:$||Ax||_2 \leq ||A||_{Frobenius}||x||_2$I'm trying to show that$||Ax||_2 \leq ||A||_{F}||x||_2$where$A$is an n by n matrix,$x\in \mathbb R^n$,$||x||_2$is the euclidean norm, and$||A||_F$is the frobenius norm. I actually wrote ... 1answer 475 views ### Proof about orthogonal subspaces There is a vector space E , which is also finite-dimensional, and it contains subspaces V1 and V2. I need help proving that: 1. ( V1∩ V2)0 = V10 + V20 2. ( V1+ V2)0 = V10 ∩ V20 Thanks! 1answer 70 views ### Proving that if two linear transformations are one-to-one and onto, then their composition is also. I am attempting to solve a problem with the following given conditions: Let V, W. and Z be vector spaces, and let$T:V \longrightarrow W$and$U: W\longrightarrow Z$be linear.Prove that if U and T ... 2answers 292 views ### Infinite direct product of fields. Let$F$be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e.$\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of$F$. ... 1answer 381 views ### Do you need the Axiom of Choice to assert that every real vector space has a norm? Math people: This question is 95% answered (the first answer) at Does every$\mathbb{R},\mathbb{C}$vector space have a norm? and Vector Spaces and AC . The questions, answers, and links found there ... 5answers 16k views ### How to solve this to find the Null Space What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{... 9answers 62k views ### Finding a unit vector perpendicular to another vector For example we have the vector 8i + 4j - 6k, how can we find a unit vector perpendicular to this vector? 6answers 1k views ### Showing 1,e^{x} and \sin{x} are linearly independent in \mathcal{C}[0,1] How do i show that f_{1}(x)=1, f_{2}(x)=e^{x} and f_{3}(x)=\sin{x} are linearly independent, as elements of the vector space, of continuous functions \mathcal{C}[0,1]. So for showing these ... 3answers 2k views ### How can a subspace have a lower dimension than its parent space? If V is a vector subspace of W, then$$\dim(V) \le \dim(W)$$Why? Does that mean that for$$W = \mathbb{R}^3\\ V = \{(0,0)\}$V$is a valid subspace of$W$? But$V$only has two ... 1answer 2k views ### Does the multiplicative identity have to be 1? I am just starting out with vector spaces and I am having a hard time understanding them. One of the requirements states that$1\mathbf{v}=\mathbf{v}$where$1$is the multiplicative identity. Does 1 ... 2answers 297 views ### How do we show every linear transformation which is not bijective is the difference of bijective linear transforms? I have been reviewing some ideas about vector spaces and came upon a surprising fact. I am not quite sure how to begin the argument because the problem requires one to construct two bijective linear ... 2answers 595 views ### Sparse basis for linear subspace Suppose I have a linear subspace of some vector space, e.g. described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is ... 3answers 3k views ### Prove that Every Vector Space Has a Basis My textbook extended the following proof to show that every vector space, including the infinite-dimensional case, has a basis. Condition:$S$is a linearly independent subset of a vector space$...
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I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
### Nullspace that spans $\mathbb{R}^n$?
My professor said that if for a $n \times n$ matrix $A$, $\text{null}(A) = \mathbb{R}^n$, then $A = 0_{n}$. Why is this true? I understand what its saying - if everything times this matrix is zero, ...
Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$. It is easy to prove that \$|S| \leq ...