Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Integral involving Matrix Exponential to solve LTI system equation

I am given that for $A$ that is $n \times n$ matrix of full rank, $$\int_{0}^{t}e^{A\sigma}d\sigma = (e^{At}-I)A^{-1}$$ Then I am using this to solve LTI system $$\dot{x}=Ax+Bu$$ Here, $x(0) = ...
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184 views

Diagonalising a matrix

I've got a matrix $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}$ and wish to diagonalise it. I find the eigenvalues as below. $$\det(A - xI) = 0 = \det\begin{bmatrix}1-x&-1\\ ...
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1answer
128 views

On the eigenvalues of the square of a real matrix $A$

I just read this snippet in a textbook "The eigenvalues of a symmetric real matrix are real (The proof follows by noting that if $A$ is symmetric, the eigenvalues of $A^TA$ are the ...
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76 views

Largest number obtained from products?

What is the largest number that can be obtained as the product of two or more positive integers that add up to 20?
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68 views

how to guess dimensions in the following subspaces

let $S$ and $T$ be two subspaces of $R^{24}$, such that $\dim(S)= 19$ and $\dim(T)= 17$, then the a. Smallest possible value of $\dim(S \cap T)$ is ? b. largest possible value of $\dim(S \cap T)$ is ...
2
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1answer
2k views

How to prove the sum of 2 linearly independent vectors is also linearly independent?

Suppose $a,b$ and $c$ are linearly independent vectors in a vector space $V$. How can I prove that $a+b$ or $b+c$ are also linearly independent?
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1answer
95 views

Is it true, there exists a non-zero integer such that $A + nB$ is invertible for $A$ is invertible and $B$ is general $3 \times 3$ matrices

If $A$ and $B$ are any $3 \times 3$ matrices and A is any invertible matrix, then there exist an integer $n$ such that $A + nB$ is invertible. It is easy to check if we take $n = 0$, then the result ...
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1answer
112 views

Proving statement about dimensions of vector spaces

While studying for my linear algebra test I came across the following problem: Let $f: \mathbb{V} \to \mathbb{W}$ be a linear transformation and let $S$ and $T$ subspaces of $\mathbb{V}$ such ...
6
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1answer
83 views

Eigen Value of a Linear Map

Let $V$ be the vector space of all continuous functions from $\mathbb{R}$ into $\mathbb{R}$ and let $T\colon V \rightarrow V$ be a linear map defined by $T(f)(x)=\int^{x}_{0}f(t)dt$. How can we prove ...
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1answer
42 views

Form of rotations in three dimensional spaces

The Euler's theorem says that for every rotation $f\in SO(3)$ of three dimensional Euclidean space there exists a orthogonal basis $e_1, e_2,e_3$ and $\theta \in [0,\pi)$ such that $$ M(f)= \left [ ...
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1answer
58 views

Eigenvalues, Eigenvectors and Eigendecomposition

If there is a symmetric matrix, say $$B = \left[\begin{array}{cc} 0 & A\\ A^T & 0 \end{array}\right]$$ where $A$ is a $m\times n$ submatrix with $m \geq n$. Is it possible to express the ...
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1answer
913 views

Linear Combinations and solutions

Let A be a 5 x 3 matrix. If $$b = a_1 + a_2 = a_2 + a_3$$ then what can you conclude about the number of solutions of the linear system Ax = b? Explain. I'm not sure about this question. All I know ...
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1answer
1k views

Covariance Matrix in Weighted Least Square Estimation

I am new to linear algebra and I have the following doubts: In weighted least square estimation of the system $Ax = b$ we minimize the weighted value of the error $e = b - Ax$ and the best $\hat{x}$ ...
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2answers
164 views

What does $\mathbb R^S$ mean when $S$ is a set? [duplicate]

Possible Duplicate: What's the meaning of a set to the power of another set? What does $\mathbb R^S$ mean when $S$ is a set? I am reading a text and I wonder if it has a special meaning ...
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1answer
131 views

How can I calculate the eigenvalue of the following matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ such that its elements are all non-negative values. I know that for any $k$, $A^k$ has elements on the diagonal which are smaller or equal to 1. Can I ...
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1answer
146 views

Give an example to show the eigenvalues can be changed when a multiply of one row is subtracted from one another

Is the following a good example? $$P=\begin{bmatrix}1&1\\1&1\end{bmatrix}$$ then, multiply the first row with 1 and subtract the first row from the second row, we can get: ...
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1answer
2k views

linear algebra proofs

I'm always having difficulties with what actually suffices as a proof, and what is obvious enough to not have to prove it. Here some I have those problems with. Let $S=\{u_1,u_2,...,u_n\}$ be a ...
3
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2answers
388 views

Proof of Cauchy-Schwarz inequality.

I'm trying to understand the proof of the Cauchy-Schwarz inequality: for two elements x and y of an inner product space we have $$\lvert \langle x,y\rangle\rvert \leq\lVert x \rVert \cdot\lVert ...
3
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1answer
275 views

is there a way to find or upper bound the largest eigenvalue of the following matrix?

I have a matrix $A \in \{0,1\}^{n \times n}$ -- i.e. a matrix with 1s and 0s only. Is there a way to find or upper bound its largest eigenvalue? I have a feeling it is related to connectivity of ...
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0answers
164 views

Positive Definite just on a Cone

Given $A \in \mathbb{R}^{n \times n}$, $C \in \mathbb{R}^{m \times n}$, and the cone $\mathcal{C}:=\{x \in \mathbb{R}^n \mid C x \geq 0\}$, find necessary and sufficient conditions on $(A,C)$ such ...
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1answer
35 views

Help with a linear algebra problem; defining a function with some properties

I'm working on the following problem: Define a linear transformation $f: \mathbb{R}^3 \to \mathbb{R}^3$ such that: $\{x \in \mathbb{R}^3: f(x) = (1, 1, 1)\} = \{x \in \mathbb{R}^3: x_1 + ...
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0answers
95 views

Generating rotations in $\mathbb{R}^n$

I want to be able to computationally generate a rotation matrix for $\mathbb{R}^n$ where $n$ might go as high as $10^4$. The naive technique would be to generate the rotation in each plane then ...
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577 views

understanding basics of linear transformations, in polynomials

$T:P_3\rightarrow \mathbb{R}$ given: $T(p)= \int_0^1x^2p(x)dx$. Prove that T is a linear transformation, and find a basis for its kernel. DO NOT SOLVE My textbook explain using only very abstract ...
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Computing very high powers of a particular Jordan block

Let $J$ be the following $k-by-k$ Jordan block: $$ J:= \begin{bmatrix} e^{i \theta} & 1 & \\ & e^{i \theta} & 1 \\ & & \ddots & \ddots \\ & & & \ddots & ...
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1answer
99 views

How to prove a determinant equality.

Suppose $u_k, v_k\in R^n$, $k=1, \ldots, m$. Define $F=\sum_{k=1}^m u_kv_k'$. How to show that $\det(I+F)=\det(\delta_{jk}+u_j'v_k)_{j,k=1}^m$? Here $I$ is the identity matrix, $u'$ is transpose of ...
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2answers
190 views

If $A$ has only one eigenvalue, what is the dimension of the corresponding eigenspace?

I'm confused about how to find the possible dimension of an eigenspace given that a matrix has exactly one eigenvalue. Suppose $A$ is a $3\times 3$ matrix, with exactly one eigenvalue $\lambda$. I ...
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3answers
733 views

What's the interpretation of a unitary matrix?

I know that a unitary matrix is a matrix whose inverse equals its conjugate transpose (or that multiplying it by its conjugate transpose yields the identity), but I don't have a deep intuition about ...
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1answer
164 views

Solution of matrix equation or matrix inequality

When there exist a positive definite solution $S$ of the following matrix equation or matrix inequality: $ SA^{T}+AS+\alpha S-\beta BB^{T}=0 $, that is, what condition on $A,B, \alpha, \beta$ can ...
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363 views

Decomposition of Matrices in Semisimple and Nilpotent Parts

For any matrix $A\in M_n(\mathbb F)$, where $\mathbb F$ is an algebraically closed field, there is a matrix $S\in M_n(\mathbb F)$ such that $$SAS^{-1}=D+N,$$ where $D$ is diagonal and $N$ nilpotent. ...
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103 views

Isomorphism question

What would be an isomorphism between $\mathcal{F}(S;V / U)$ and $\mathcal{F}(S;V)/ \mathcal{F}(S;U)$, where $S$ is a set, $V$ a vector space and $U$ a subspace of $V$. $\mathcal{F}(A,B)$ denotes the ...
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1answer
77 views

$\sum_{i=1}^n \frac{p(a_i)}{q'(a_i)}=p_{a_{n-1}}$

In a question in Fuhrmann's "A polynomial approach to linear algebra" it is stated that $$\sum_{i=1}^n \frac{p(a_i)}{q'(a_i)}=p_{a_{n-1}},$$ where $p,q$ are polynomials over a field with $deg(p)=n-1$ ...
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Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
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2answers
64 views

Verifying a bijection

Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets ...
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2answers
80 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
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1answer
270 views

Proving matrices $W_1$ and $W_2$ are subspaces and finding their dimensions

Let $V = M^{2\times 2}(\bf F),$ $$W_1 =\left\{\begin{bmatrix}a & b \\c & a\end{bmatrix}\in V\;:\; a, b, c\in F\right\}$$ and $$W_2 =\left\{ \begin{bmatrix}0 & a \\-a & ...
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1answer
91 views

Plotting temperature over time excel

I am doing an uni assignment and have worked out a linear equation which plots temperature over time. I have this in a graph now but that required me to use a lot of calculations in the spreadsheet. ...
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1answer
2k views

Basis for upper triangular matrices of $M_n(F)$

I'm asked to find a basis for $W$, which is a subspace of $M_n(F)$. $W$ is the subspace containing all upper triangular $n \times n$ matrices. ...
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Can I use the Gram-Schmidt procedure to generate general solutions to a linear ODE?

Is this possible? The question popped into my head as my ODE instructor was teaching us how to solve second-order linear ODEs. Anyway, he said the following things: A second-order linear ODE has ...
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3answers
264 views

Intuitively (concretely actually) what happens when you multiply a matrix by its transpose?

The construct $A^TA$ for $A$ any $m \times n$ matrix seems to appear often in formulae and results. For example I was reading that square root of eigenvalues of $A^TA$ (an $n \times n$ matrix) are ...
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2answers
93 views

Stopping after 4 vectors, when looking for a basis of $\mathbb{R}^5$

I'm asking in regards to Question 8 here: http://www.math.ucla.edu/~jheez/hw2solutions.pdf The answer given says that you can stop searching once you've identified four vectors that are a basis to ...
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243 views

Homogeneous Linear Transformation

This is a general question but can someone provide a worked example of a 3d transformation? Or a link that has a worked example of one? I've looked on the internet for a long time and couldn't find ...
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2answers
51 views

Serie of a matrix

Let $A$ a matrix $n\times n$. Define $e^A=\sum ^{\infty}_{n=0} \frac{A^n}{n!}$ (also you can see this question). If $A$ is a diagonalizable matrix, find $e^A$ in terms of eigenvalues of $A$. I was ...
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3answers
108 views

Determine whether $\{(x_1,x_2,x_3)^T \mid x_1 = x_2 = x_3\}$ is a subspace

Determine whether $\{(x_1,x_2,x_3)^T \mid x_1 = x_2 = x_3\}$ is a subspace I know that I must show that it is closed under addition and multiplication, but I'm confused as to how I should ...
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1answer
618 views

Oblique projection matrix - proof definition is correct.

The oblique projection matrix on $\text{range}(X)$ orthogonal to $\text{range}(Y)$ is given by $P = X (Y^\top X)^\dagger Y^\top$. Prove that the above definition is right, i.e. it holds that ...
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2answers
64 views

Commutative matrices confusion

I am confused of some commutative properties of some matrices, so here is the question. What would constitute(or be the name of) a matrix that is always commutative? Which matrices would satisfy this ...
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1answer
57 views

Minimizing the product of some variables with constant summation having an additional condition

What is the minimum of $a_1\times a_2 \times \dots \times a_n$ such that $a_1+a_2+\dots+a_n=S$ and $0 < x \le a_i \le (1+\alpha)\frac{S}{n}$? My conjecture is that we need to set as many ...
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64 views

Finding Representation Matrix

Define $\alpha$ as $\alpha(p)(x)=(x-1)(p'(x)+p'(1))$ Let V also be the space of polynomials of degree less than or equal to 2. The basis given is ${{1,x,x^{2}}}$ After after what I hope to have been ...
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2answers
53 views

Can you help with the spaces generated?

Let A be a $n \times n$ matrix and $[A_{1},...,A_{n}]$ The columns of the matrix, my question is: Why if $Ax=b$ then $A^{T}b\in Span [A_ {1}, ..., A_ {n}]$? Where x is any vector of length n ...
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1answer
102 views

Finding vector space complements

Find a vector space complement to $\mathbb{R}(1 + x + x^2) + \mathbb{R}(x - x^2 - x^3)$ in $\mathbb{R}[x]\le_3$. (This is referring to the set of polynomials with degree less than or equal to 3.) I ...
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1answer
65 views

Does this conic combination generate all $n\times n$ real symmetric positive-semidefinite matrices?

Let $e_i$ denote a column-vector of length $n$ whose entries are all zero except for the $i$-th entry that is 1. Now consider the set of $n\times n$ matrices given by $$\mathcal{M}_n=\left\lbrace ...