Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

learn more… | top users | synonyms

1
vote
1answer
28 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
1
vote
1answer
22 views

Gram-Schmidt of 2 by 2 matrix

Given $A=\begin{bmatrix} 2 &-1 \\ -1 & 2 \end{bmatrix}$, I take the first column of $A$, and divide it by its norm to find $q_{1}=\begin{bmatrix}\frac{2}{\sqrt{5}} \\ ...
1
vote
4answers
173 views

What values of a is the set of vectors linearly dependent?

The question is is "determine conditions on the scalars so that the set of vectors is linearly dependent". $$ v_1 = \begin{bmatrix} 1 \\ 2 \\ 1\\ \end{bmatrix}, v_2 = \begin{bmatrix} 1 \\ a \\ 3 ...
1
vote
2answers
40 views

How would you determine the transformation matrix?

Suppose there exists a linear transformation $T$ where $T: \mathbb{R^3} \to \mathbb{R^5}$ and $T(\textbf{x}) = \text{A} \textbf{x}$. Given $$ \text{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = ...
1
vote
2answers
44 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
1
vote
2answers
569 views

show projection matrix is equal to matrix times its transpose

Let $V$ be an $n$-dimensional real inner product space and let $a=\lbrace v_1,v_2,\dots v_n \rbrace$ be an orthonormal basis for $V$. Let $W$ be a subspace of $V$ with orthonormal basis $B = \lbrace ...
1
vote
1answer
36 views

If $S$ is an isometry, why is $\sqrt{S^{*}S}$ a positive and hence self adjoint operator?

I am trying to show that $S$ being an isometry leads to the fact that all singular values of $S$ equal 1. I know a key part of the proof is showing that $\sqrt{S^{*}S}$ is self adjoint so that I can ...
1
vote
2answers
40 views

If $R$ is a positive operator and $R^2 = T^{*}T$, why does this mean we can write $R$ as $\sqrt{T^{*}T}$?

If $R$ is a positive operator and $R^2 = T^{*}T$, why does this mean we can write $R$ as $\sqrt{T^{*}T}$? Should I be thinking of the square root of an operator as in the way I think about it when it ...
1
vote
3answers
35 views

Positive-definitive matrix: Confusion about “vector of zeros”

According to Wikipedia: "In linear algebra, a symmetric $n \times n$ real matrix $M$ is said to be positive definite if $z^TMz$ is positive for every non-zero column vector $z$ of $n$ real ...
1
vote
2answers
46 views

Question about eigenvalues and eigenvectors

If A is n by n matrix of complex numbers, then it's easy to show that if c is an eigenvalue of A, then c^2 is an eigenvalue of A^2, like.. is v is an eigenvector of A corresponding to the eigenvalue ...
1
vote
1answer
153 views

Understanding a Certain Proof of the Cauchy-Schwarz Inequality

Cauchy-Schwarz Proof Outline: Let $u,v \in V$ s.t. $u,v \ne 0$. Then $0 \le \| u - rv\|^2$. Let $r = {\overline{\left\langle u,v\right\rangle} \over \left\langle v,v \right\rangle}$. Then one ...
1
vote
2answers
52 views

QR Factorization: Size of Matrix R?

So I'm reading about QR factorization. Here's what my book says: If M is an n x m matrix with linearly independent columns, then there is an n x m matrix Q whose columns are orthonormal and an upper ...
1
vote
2answers
70 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
1
vote
2answers
40 views

Eigenvalues of rectangular matrices

If we have a non-zero real $n$ by $m$ matrix $M$, then there may exist a non-zero unit vector $v$ of $m$ elements so that $Mv = 0$. I understand we can't call this an eigenvector with eigenvalue $0$. ...
1
vote
1answer
55 views

Symmetric bilinear forms in characteristic 2

This is a homework question: Prove that: In field $K$ of characteristic $2$, for symmetric bilinear forms on $K^2$, there exist a basis where the matrix of the bilinear form is either diagonal or ...
1
vote
1answer
54 views

How to show the pushforward is linear using equivalence classes of curves?

Let $M$ be a $C^k$ manifold of dimension $n$. I've constructed the tangent space at $a \in M$ as follows: first I've introduced the following equivalence relation in the set of maps $\gamma : ...
1
vote
1answer
44 views

Matrix determinant operations.

Suppose you are trying to find the determinant of the following matrix using the "upper triangulation" method: $\begin{matrix} 1&0&0\\ 0&1&0\\ 1&1&1 \end{matrix}$ If I take ...
1
vote
2answers
1k views

Easiest way to solve system of linear equations involving singular matrix

I am trying to balance an unbalanced chemical equation by using setting up a system of linear equations to solve for the stoichiometric coefficients in the chemical equation. After setting up a ...
1
vote
2answers
42 views

Let $V$ be a vector space over a field $\mathbb F$. Show $0 \in \mathbb F$ is the only element : $\alpha v = 0 \in V, \alpha \in \mathbb F, v \neq 0$

Let $V$ be a vector space over a field $\mathbb F$. Show $0 \in \mathbb F$ is the only element : $\alpha v = 0 \in V, \alpha \in \mathbb F, v \neq 0$. Is this statement true ? I know in a field ...
1
vote
1answer
97 views

How can I show that the Hadamard product of REAL matrices is positive?

By Hadamard product, I mean the componentwise product of positive real matrices. By positive, I mean that it is positive semidefinite. I am currently trying to show this but am not sure if it is ...
1
vote
4answers
82 views

Linear maps using Tensor Product

While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation: "To give a linear map $V ...
1
vote
1answer
211 views

$2\times 2$ matrices over complex numbers

I am trying to solve this problem. If $A$ is a $2 \times 2$ matrix with complex entries, then $A$ is similar over $\Bbb C$ to a matrix of one of the two types $$ M= \left[ {\begin{array}{cc} a ...
1
vote
1answer
65 views

Question regarding Sheldon Axler's proof that every operator on a complex vector space has an eigenvalue

For reference, Axler's proof was copied in full in this question. In the beginning of that proof, Axler chooses the nonzero vector $v$ without loss of generality. This made me doubt his proof ...
1
vote
1answer
173 views

How to find a positive semi-definite linear combination?

Suppose we are given two explicit symmetric matrices $X$ and $Y$ and we'd like to find a non-zero real linear combination $aX+bY$ that is positive semi-definite (if possible). Is there a way to go ...
1
vote
2answers
489 views

Find bases for subspaces spanned by vectors.

The standard basis for $P_2(\mathbb R)$, the vector space of quadratic polynomials of the form $ax^2+bx+c$ is the set $S=\{1,x,x^2\}$. Find bases for the subspaces of $P_2(\mathbb R)$ spanned by the ...
1
vote
3answers
458 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
1
vote
2answers
206 views

Linear Algebra need help with proof please over eigenspaces

I know that if x and y are distinct eigenvalues of an nxn matrix A, then the intersection of eigenspaces is the 0 vector. How can I prove this?
1
vote
2answers
77 views

Linear Independence for Column Vectors?

Here's the problem I'm struggling with: Let $A$ be an $m \times n$ matrix and let $B$ be an $n \times m$ matrix ($n \neq m$). We are given that $AB = I_m$. Are the column vectors of $A$ linearly ...
1
vote
3answers
62 views

Finding the basis of a vector space out of matrices

I am to find a basis for the vector space $M$ formed by all $(n \times n)$-matrices. Now, I am finding this to be quite different from previous exercises with bases, where I only have had to construct ...
1
vote
1answer
80 views

Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
1
vote
2answers
84 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
1
vote
2answers
771 views

Unique decomposition of a vector space into a direct sum

Suppose I have a vector space W that is the direct product of two subspaces, U and V. So: $W=U\oplus V$ My working definition of direct product is that $W = U + V$ and $U\cap V = 0$. Now my problem ...
1
vote
1answer
22 views

Need help in finding a basis for a subspace defined by a function

I've been given a set defined as follow: set W = {x(t): x(t) = c1cos(bt)+c2sin(bt), where c1,c2 are arbitrary constants, and b is a fixed constant} I have to find a basis for that set and justify why ...
1
vote
2answers
61 views

Is this map an isomorphism?

Let $f : M_{2 \times 2} \to \Bbb{R}$ be given by $$ \{ \{ a, b \}, \{ c, d \} \} \mapsto ad-bc $$ To prove something is an isomorphism it has to be 1-1, onto and preserve structure. Can someone ...
1
vote
1answer
81 views

Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
1
vote
1answer
49 views

How to disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$?

How can I disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$? My approach is to use the parallelogram law to show that if I have two vectors $u,w ...
1
vote
1answer
62 views

Different basis over $\Bbb{R}$ and $\Bbb{C}$

V is a finite dimensional vector space over $\Bbb{C}$ and {v$_1$,...,v$_n$} be a basis of V. Show {v$_1$,iv$_1$,...,v$_n$,iv$_n$} is a basis of V over $\Bbb{R}$ and conclude: ...
1
vote
1answer
210 views

How does this proof of the Cauchy-Schwarz Inequality work?

I was watching a Khan Academy video on the Cauchy-Schwarz Inequality, and I just can't seem to understand the proof, and the comments on the video don't seem to help. The video is here. First, he ...
1
vote
2answers
145 views

Show that a set of vectors spans $\Bbb R^3$?

Let $ S = \{ (1,1,0), (0,1,1), (1,0,1) \} \subset \Bbb R^3 .$ a) Show that S spans $\Bbb R^3$ b) Show that S is a basis for $\Bbb R^3 $ I cannot use the rank-dimension method for (a). Is it ...
1
vote
1answer
252 views

What commutes with a matrix in Jordan canonical form?

The question I would like answered is the following: Given a matrix $G$ and that $G$ commutes with another matrix $X$, that is $[G, X] = 0$, what is $X$? Or more generally, what properties of $X$ may ...
1
vote
2answers
111 views

Determining Linear Independence and Linear Dependence?

I understand that when solving for a linear dep/independent matrix, you can take the determinant of the matrix and if it is zero, then it is linearly dependent. However, how can I go about doing this ...
1
vote
1answer
40 views

An equation related to covariance matrix, square root of the matrix, and Euclidean norm.

How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which $\Sigma $ is a covariance matrix. I tried some numerical examples in ...
1
vote
1answer
33 views

Linear Operator identity prrof [closed]

Let A,B be invertible linear operators. Prove the identity: $B^{-1}-A^{-1}=B^{-1}(A-B)A^{-1}$
1
vote
2answers
94 views

Linear Transformations from $\mathbb{K}^n$ to $\mathbb{K}^m$

I'm studying linear functionals and dual spaces... And I found this exercise: Let $f_1,f_2,...f_n \in (\mathbb{K}^n)^*$. For each $\alpha \in \mathbb{K}^n$, define: $$T(\alpha)=(f_1(\alpha), ...
1
vote
1answer
186 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
1
vote
1answer
47 views

Subspaces of finitely-generated vector space

Let $W_1,W_2,W_3$ be linear subspaces of finitely generated vector space V. Assume that $W_1\cap W_2=W_1\cap W_3=W_2 \cap W_3=\{\theta \}$. Is $W_1+W_2+W_3$ is always direct sum? I think that ...
1
vote
2answers
36 views

Proof: linearly dependence for particular set in $\mathbb{R}^4$

The problem that I can't seem to solve is: Let $S$ be the set of all vectors in $\mathbb{R}^4$ with exactly $2$ entries equal to $1$ and all the rest of its entries equal to $0$. Is $S$ linearly ...
1
vote
3answers
52 views

A question on inequalities

What is the solution set of the inequality $$ \frac{2x - 1 }{x+1}\lt0$$ One answer that is quite simple to get is $$x\lt1/2 $$ What can be the other value for the solution set...??
1
vote
1answer
83 views

Linear Algebra Span question

Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$. Since they all fall on the same plane, I can't seem to find a ...
1
vote
3answers
37 views

Showing set linear independence

How do I show that the set $\{ e^x , ... ,e^{nx} \}$ is linearly independent? I tried using induction as the base case of $\{ e^x \}$ and even $\{ e^x, e^{2x} \}$ is easy, but I can't use the I.H. ...