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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Why is $\det(A-\lambda I)=(\lambda-c)^n$ when $(A-cI)^n=0$?

Let $A$ be a $n\times n$ matrix and suppose that $(A-cI)^n=0$ for some scalar $c$. Then why the characteristic polynomial of $A$ is $(x-c)^n$?
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2answers
13 views

Compute $B=QAQ^{-1}$

$A,B$ are $n\times n$ matrices, $B=QAQ^{-1}$, and I know $A$ and $B$, how to compute $Q$? I know if $T$ a linear transformation, and with different basis we get $A$ and $B$, and we could use these ...
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0answers
6 views

Number of pivot columns in a 4x6 matrix for spanning set to occur

How many pivot columns must a 4x6 matrix have if its columns span $\mathbf{R}^4$? Explain. So, in my head, this is pretty clear: You need four dimensions => So you need a minimum of four vectors that ...
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1answer
14 views

Square Root of an Inverse Matrix

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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0answers
7 views

Linear Algebra Analytical Exercise

This one has me stumped... $$H=C(sI-A)^{-1}B$$ and $$H_{CL} = C(sI-A+BK)^{-1}BG$$ Show that $$H_{CL} = H[I+K(sI-A))^{-1}B]^{-1}G$$ Any hints would be greatly appreciated!
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Find a basis for symmetric $2 \times 2$ matrices [on hold]

Find a basis for the space of all $2 \times 2$ symmetric matrices. I do not even know how to start. please explain it to me step by step
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1answer
9 views

Solving symbolic linear equations with maple

How can I solve linear equations of the following type in Maple? $$\begin{pmatrix} 1 & 1 & 1 & 1\\ b-c & c-b & a-b &0 \\ b-d & d-a & 0 &a-b \end{pmatrix} ...
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0answers
4 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
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0answers
18 views

The space of symmetric 2x2 matrices? Find the basis of the matrix [on hold]

The space of symmetric 2x2 matrices? please help me with this question. I think this question is asking to find basis of the matrix
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1answer
20 views

Simple Eigenvalue finding question (by gauss elimination)

I saw a method for finding eigenvalues by using Gauss elimination to find an upper triangular matrix, then just taking the diagonal elements as the eigenvalues. It seems to work except for this case: ...
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0answers
13 views

Subspace vector proofs problem [on hold]

I'm having trouble understanding/solving this proof. QUESTION: Prove the set P_3 is a subspace of P_4 with standard operations, where P_n is a vector space of all polynomial functions with degree n ...
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1answer
20 views

Basis for vector space $\mathbb{R}^{m\times n}$

My question is whether my solution to the following problem is valid. The problem is from Artin's Algebra, chapter 3: Let $(X_1,\cdots,X_m)$ and $(Y_1,\cdots,Y_n)$ be bases for $\mathbb{R}^m$ and ...
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1answer
20 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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2answers
31 views

What does determinant of linear operator mean?

I am solving problem (Linear Algebra by Hoffman, Excercise 5.4.8) : Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $B$ be a fixed element of $V$ and let $T_B$ be the ...
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2answers
14 views

Representation of Matrix with Rank 1

Prove that every $m \times n$ matrix of rank $1$ has the form $A=XY^t$, where $X,Y$ are $m$- and $n$-dimensional column vectors. How uniquely determined are these vectors$?$ My attempt: I thought ...
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0answers
29 views

Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...
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1answer
27 views

True or False Question About Linear Algebra

I 'm new in Stack and I need help with a few questions about linear algebra. I'm trying it but I cannot. TRUE OR FALSE 1) Let $A,B$ and $C$ be $nxn$ matrices such that $C$ is invertible and $B=C.A.{ ...
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1answer
13 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
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1answer
10 views

Linearization of a function in the point 0, 0

The linearization of the function $ f(x, y) = 1 + 2(x + 1) + 3(y + 1) + 4x^2 + 5y^2 $ in the point (0, 0) is given by: $ L(x, y) = 6 + 2x + 3y $ I know this is true, but how does one come to this ...
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1answer
16 views

Range and Nullspace of a transformation

Let $$(Tf)(x)=\int_0^xf(t)dt$$ be a transformation from the vector space V of all functions from $\mathbb{R}$ to $\mathbb{R}$ which are continuous. Describe the range and nullspace $T$. To me it ...
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0answers
19 views

Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
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1answer
21 views

There are at least four 3-dimensional subspace in $R^4$?

Shouldn't there be EXACTLY four 3-dimensional subspaces in $R^4$? My reasoning is that 3-d subspaces occur in $R^4$ when the rank of the augmented matrix of {$c_1:c_2:c_3:c_4$} is 3?
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1answer
21 views

What does $b^*$ mean?

What is this notation, my book explains nothing of it. I've colored it in yellow! I am guessing it stands for $b^{-1}$ or $b^1$?
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3answers
29 views

Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$? [duplicate]

Basically does a vector set that is linearly independent in $R^n$ automatically span $R^n$? My initial thought is yes, but is there some counterexample that can disprove this?
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0answers
9 views

Markov chains: identifying a nonregular transitional matrix

I am currently TA'ing for a course in which the students are soon to learn about Markov chains and stochastic matrices. During the sections, it refers to the possible existence of a stable state and ...
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1answer
19 views

Prove that the determinant of this matrix is non-zero.

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 ...
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1answer
15 views

Some operation like determinant

we have determinant operation that is like below: $ det(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}) $= $ (-1)^{1+1}a(ei-fh)+ ...
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1answer
25 views

Every basis in the space of matrices 2x2 contains a non invertible matrix?

My initial thought that this was true; however, I thought of this counter example that may be possible? Would something like: ...
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0answers
22 views

Finding an upper bound for solution of $Ax=b$

Is there any upper bound on $x$ in $ Ax=b $ using some features of $A$, for example $ \min(\lambda_i) $ or $ \max(\lambda_i) $ eigenvalues of $A$ ? I've tried to find something like $$ \Vert x\Vert ...
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2answers
10 views

find the intersecting line in the provided planes $( 3x+2y+z = -1 ; 2x-y+4z=5)$

find the intersecting line in the provided planes $(3x+2y+z = -1 ; 2x-y+4z=5)$. I keep getting the wrong answer, here is my approach: I set it up in an augmented matrix: $$ ...
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1answer
9 views

Find an orthonormal basis for W and $W^{\perp}$

Consider $\mathbb{C}^3$ with the standard inner product (that is, the dot product), and let $W = \text{span} \{(1, 0, 1), (i, i, i)\}$. (a) Find an orthonormal basis for $W$. (b) Find an orthonormal ...
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1answer
9 views

Statement about non-homogeneous linear system with $n+1$ equations and $n$ unknowns

Let $M$ be some non-homogeneous linear system of $m$ equations and $n$ unknowns and $m=n+1$. Is it true that if the row echelon form of the augmented matrix (extended coefficient matrix) of $M$ ...
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2answers
27 views

Determine the values of c for which the equation Ax = b is consistent.

Determine the values of c for which the equation Ax = b is consistent. A= ...
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1answer
22 views

Concerning crossproduct and orthonormality of vectors

If you have 3 orthonormal vectors, $\pmatrix{u_1 \\ u_2 \\ u_3}$, $\pmatrix{v_1 \\ v_2 \\ v_3}$, $\pmatrix{w_1 \\ w_2 \\ w_3}$ such that $u= v \times w$ (crossproduct) can it be true that $v_i v_j + ...
2
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3answers
185 views

Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...
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1answer
8 views

Find vectors $(a,b)$ and $(c,d)$ so that the frame $\{(1,1),(1,-1),(c,d),(a,b)\}$ is PRR equivalent to the frame

Find vectors $(a,b)$ and $(c,d)$ so that the frame $\{(1,1),(1,-1),(c,d),(a,b)\}$ is PRR equivalent to the frame ...
2
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2answers
30 views

Linear Algebra Subspace question

Here is the following question and answer for the question. I don't seem to quite grasp the answers or how the answers are what they are. Requirements for subspace: the zero vector is in the ...
2
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3answers
29 views

Relation between trace and rank for projection matrices

If $A $ is an $n \times n$ matrix over $\mathbb C$ such that $A^2=A$ then is it true that $\operatorname{trace} A = \operatorname{rank} A$?
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1answer
11 views

Is the pair controllable/observable?

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and ...
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5answers
39 views

Whether a matrix is a zero matrix

If a real square matrix $A $ is similar to a diagonal matrix and satisfies $A^n=0$ for some $n\in \mathbb N $,then can it be proved that $A$ must be a zero matrix?
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1answer
26 views

Bases and Matricies

Assume $ dim V = 2 $ and $\{e_1, e_2\}$ is a basis of $V$. Suppose that $$ M(T,\{e_1,e_2\}) = \left (\begin{array}{cc} 0 & 0 \\ 1 & 1 \end{array} \right) $$ Find a basis $\{v_1,v_2\}$ of $V$ ...
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0answers
21 views

Tensor product of two linear map and its matrix representation

Suppose $T_1: \mathbb{R}^n\to\mathbb{R}^n$ be any linear map and wrt a basis $\{e_1,\dots,e_n\}$ the matrix of $T_1$ is $M$, aand $T_2:\mathbb{R}^m\to\mathbb{R}^m$ be another linear map whose matrix ...
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1answer
50 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
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1answer
19 views

Problem on Finding the rank from a Matrix which has a variable

$$ A = \begin{bmatrix} 1 & -1 & -2 & -3 \\ -2 & 1 & 7 & 2 \\ -3 & 3 & 6 & \alpha \\ 7 & -6 & -17 & -17 \end{bmatrix} $$ Find the rank when ...
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1answer
44 views

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$?

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$ ? I've been trying to sketch a proof by induction, but it seems more complicated that it should ...
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0answers
9 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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1answer
8 views

Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then ...
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1answer
16 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
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0answers
19 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
2
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1answer
28 views

Rank of a matrix and dimension of the image

I'm teaching linear algebra to first year students, and I was recently asked why is the rank of a matrix, representing a linear application in a given basis, equal to the dimension of the image space ...