# 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 ...

74 views

### Prove that N(T)=0 and R(S)=U

Let $T:U \to V$; $S:V \to U$ and $ST:U \to U$. Prove that $N(T)=\{0\}$ and $R(S)=U$. My professor gave us a fact at some point that if $ST=ID(U)$ we have S is surjective and T is injective. I am not ...
35 views

### What is the intuition behind Gramian method for linear independence? and Is there $simple$ proof of it?

I'm trying to figure out the intuition behind Gramian method to determine the linear independence of functions. I searched the web for such simple intuitive explanation and found nothing. I tried ...
53 views

### Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
46 views

### Basis for a subspace

I need to calculate the basis for $$W = \lbrace (a,b,c,d) \: : \: a+b+c = 0 \rbrace.$$ I find it hard to understand how does the fact that d is not part of the equation effects the basis. Thanks ...
44 views

### Diagonalizable by orthonormal matrix

Given the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ Explain why $A$ can be diagonalized by an orthonormal matrix and find an ...
48 views

34 views

### On the dimension of subspaces of the vector space given by the product of polynomials.

I was asked this question orally so feel free to also correct how the question is written. Given the vector space of polynomials in the variable $x$ with degree $\le 4$ and the vector space of ...
Let $M$ be a subspace of $\mathbb R^4$ which is spanned by the vectors $v_1 = (1,0,-1,1)$ , $v_2=(0,1,2,1)$. Find the orthogonal complement $M^T$ of $M$ and the orthogonal projections of the vector $v=... 1answer 29 views ### Show that$\operatorname{Span}(C) = V_1 \cap V_2$Let$\mathbb{R}[x]$be the set of polynomials, and let $$V_1 = \{a_1x + a_2x^3 + a_3x^5 \mid a_1, a_2, a_3 \in \mathbb{R}\}$$ $$V_2 = \{b_1x^2 + b_2x^3 + b_3x^4 \mid b_1, b_2, b_3 \in \... 1answer 24 views ### Prove that T(u) is linearly independent in W Let V and W be two vector spaces over \mathbb{R} Suppose X \subseteq V is a nonempty linearly independent set and T:V \rightarrow W is an injective linear map. Prove that {T(u): u \in X} ... 1answer 40 views ### Precedence of operations in vector spaces Suppose that V is a vector space over \mathbb R (for simplicity) with addition denoted by \oplus and scalar multiplication denoted by \otimes. Let \mathbf u, \mathbf w \in V and let \lambda ... 1answer 35 views ### Linear Transformation Basis Exercise I have tried to solve the following exercise. Is it right? Consider the linear transformation L: ℝ⁴→ ℝ³. Knowing that:$$ L \begin{pmatrix}2\\0\\0\\0\end{pmatrix} = \begin{pmatrix}2\\2\\2\end{... 1answer 24 views ### Linear transformation explanation I have the following exercise: Consider the linear transformation L: ℝ³→ ℝ². Knowing that: $$L \begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix} \space\space\space\space\... 0answers 19 views ### Prove that (f, g, h) is a linearly independent list of vectors in \mathbb{R}[x]^S "Recall that \mathbb{R}[x] is a vector space. Suppose that f, g, h \in \mathbb{R}[x]^S and that there is q \in S such that f(q) = 1, g(q) = x^2 + 1, and h(q) = x^2 + x. Prove that (f, g, ... 0answers 21 views ### Is it better to average the log2 for a series of numbers or just the numbers themselves? And, how would you test or prove this? Lets say I'm trying to compare two vectors for similarity and normalizing them before hand based on some mean or standard deviation combo for the purpose of finding the similarity between the 2 ... 1answer 62 views ### Differentiation w.r.t. the \mbox{vec} operator I am stuck at solving the following derivative$$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$where X is an m \times n matrix and \mbox{vec} is the vector/stack operator. I have tried using ... 3answers 27 views ### Unit vector c perpendicular [closed] Find a unit vector c perpendicular to both of the vectors a = 0j + 1j - k and b = 2i + 2j – k. Just need steps/hints or even the solution would help me check if I go it right. 1answer 38 views ### Construct a linear map M : V → V with the property that K = \{v ∈ V\mid Mv = 0\}. "Suppose that V is a vector space and L : V → V is a linear map. (i) Let K ⊂ V be the set of all vectors v ∈ V such that L(v) = −v. Show that K is a subspace of V . (ii) Construct a linear ... 1answer 31 views ### Clarification of ideas concerning a quotient space. Suppose I have a vector space V, and I identify x\in V with \lambda x\in V, where x\neq 0 and \lambda>0, \lambda\in\mathbb{R}. I'm confused about two things: (1) Can I define a norm on ... 1answer 20 views ### Definition of complex vector space from Rudin RCA This is definition of complex vector space from Rudin's book. He write that to each pair (\alpha,x), where x\in V and \alpha is scalar there is associated a vector \alpha x\in V. That's right. ... 3answers 60 views ### Why a linear trasformation doesn't depend on the bases we choose Imagine we are given the following linear transformation: f(x,y) = (x+y, x) Imagine we choose a base, let's call it B_{1} and we apply the function to some vector. Now imagine we choose another ... 0answers 14 views ### Show that V1 is a linear subspace of R[x]? "Let \mathbb{R} be the set of polynomials, and let V_1 = (a_1x + a_2x^3 + a_3x^5 | a_1, a_2, a_3 \in \mathbb{R} ) and V_2 = (b_1x^2 + b_2x^3 + b_3x^4 | b_1, b_2, b_3 \in \mathbb{R} ) be ... 1answer 24 views ### Understanding components of a vector I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector \vec A in the ... 2answers 26 views ### Show that V = \ker T \oplus \operatorname{im}T where T is an idempotent linear operator [duplicate] I have to prove that if T is an idempotent (T^2=T) linear operator then space V = \ker T\oplus\operatorname{im}T. My first try was to think about the basis of subspace \ker T. Let say (e_1,... 1answer 33 views ### Non-negative Linear Span of Vectors I would like to understand if there is a common concept of a `linear span' of a set of vectors which are combined with non-negative multipliers. I know that usual definition of the span of a set of ... 4answers 62 views ### How to prove a W=\{(x,y):3x+y=0\} is a subspace of R^2 How can I prove this vector W is a subspace of \mathbb{R}^2 (closed under addition and scalar multiplication) if I have the condition 3x+y=0. Does this mean this vector already has the 0 ... 4answers 50 views ### Direct sum of vector subspaces equals \mathbb R^3 I tried solving the following linear algebra problem, I hope that someone can tell me if this is a good solution, and if not, how I should solve it. Let U and W be vector subspaces of the vector ... 0answers 37 views ### How to fill the basis of vector subspace up to \mathbb{R}^3? So, if we're given a vector subspace V of \mathbb{R}^3 with basis:$$B_V=\{(-1,1,1),(2,1,-1)\}$$How can we find a basis of vector subspace F such that:$$V\oplus F=\mathbb{R}^3 \ ?$$What I did ... 1answer 26 views ### what is the dimension? The dimension of the row space of a 8\times 8 matrix A is 5.if \mathbb{R}^{8\times 10} is a vector space of 8 \times 10 matrices with real entries. Then S_{A} = \{ B \in \mathbb{R}^{8\times ... 0answers 43 views ### Action of a Linear Functional on a Polynomial I was hoping to find a good canonical reference for the mathematics behind something called the action of a linear functional L on a polynomial p(x) which is denoted \langle L|p(x)\rangle ... 0answers 36 views ### What operator has these algebraic properties? I am working in a space V of objects that behaves like a vector space with a partial ordering \preceq. I have discovered an operator f:V\times V \rightarrow V with the following properties: For ... 2answers 22 views ### Subspace proof wording not sure how to word my answer for this question: Let V be a vector space and let H and K be two subspaces of V. Show that the following set W is a subspace of V: W={u+v: u ∈ H, v ∈ K} I'm pretty ... 0answers 30 views ### Linear Algebra Axler, 3e, Exercise 1c, P11 P11. Prove that the intersection of every collection of subspaces of V is a subspace of V. My solution was very similar to the one from linearalgebras.com -- Solution: Assume U_i are subspaces of ... 1answer 36 views ### What's the difference between V\times W and V\otimes W where V and W are vector space? Let V,W vector spaces. I don't really understand what is V\otimes W. To me it looks the same that V\times W. Do you have any explanation ? 3answers 42 views ### Set of orthogonal vectors in \mathbb{R}^n How can we show that a set of pairwise orthogonal vectors in \mathbb{R}^n has size at most n? I know it seems very intuitive, but not sure what the formal proof would look like (whether "... 0answers 21 views ### when can i move a sum through this tensor product? If I have a vector space V^{(1)}\otimes V^{(2)} and I have some ray \sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k, is the only solution that s_k = s \forall k? All x_k... 2answers 33 views ### vector and curl identity This popped up in my notes and the author made no remarks about the properties used \bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0} Then, \... 0answers 26 views ### Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem Let X be a real or complex vector space, and let p \colon X \to \mathbb{R} be a real-valued function satisfying$$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$and$$p(\alpha x) = \... 1answer 50 views ### Dimension of set of$3\times 3$matrices? Calculate the dimension of the image and kernel of each linear transformation. (Hint: you do not need to find a matrix representing the linear transformation.)$(a)P\colon\Bbb R^3\to\Bbb R^3$by (... 1answer 52 views ### Subspace proof$\{\,f \colon\Bbb R \to\Bbb R \mid f(x + 1) = f(x) + 1\,\}$I have no idea how to show that this is a subspace. Isn't$f(x)=x$and$f(x)=3x$a counter-example? It is not closed under scalar multiplication? But I guess it is..$[e]$forgot to say that the ... 1answer 51 views ### A question about diagonalizable linear operator Suppose$T$is a diagonalizable linear operator on a finite dimensional vector space$V$. Prove$V$is T-cyclic subspace of itself iff every characteristic subspace of it is one-dimensional. It ... 1answer 34 views ### switching between P3 and R4 vector spaces first of all I couldn't find a better way of describing what I really meant so here it goes. Lets say I have a vector space$\mathbb{P}_3[\mathbb{R}]$and a sub vector space$U = \textrm{span}\{ x^2 +...
How does one generate these polynomials using the gram-Schmidt algorithm? I know how it should work, but I get 0 as the value for the scalar product of (p1,q0) and q1 should be 2x not x. q1\left(x\...